Theorem mhphf 39775

Description: A homogeneous polynomial defines a homogeneous function. Equivalently, an algebraic form is a homogeneous function. (An algebraic form is the function corresponding to a homogeneous polynomial, which in this case is the (𝑄‘𝑋) which corresponds to 𝑋). (Contributed by SN, 28-Jul-2024.)

Hypotheses

Ref	Expression
mhphf.q	⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
mhphf.h	⊢ 𝐻 = (𝐼 mHomP 𝑈)
mhphf.u	⊢ 𝑈 = (𝑆 ↾s 𝑅)
mhphf.k	⊢ 𝐾 = (Base‘𝑆)
mhphf.m	⊢ · = (.r‘𝑆)
mhphf.e	⊢ ↑ = (.g‘(mulGrp‘𝑆))
mhphf.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
mhphf.s	⊢ (𝜑 → 𝑆 ∈ CRing)
mhphf.r	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
mhphf.l	⊢ (𝜑 → 𝐿 ∈ 𝑅)
mhphf.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)
mhphf.x	⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁))
mhphf.a	⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))

Assertion

Ref	Expression
mhphf	⊢ (𝜑 → ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))

Proof of Theorem mhphf

Dummy variables 𝑎 𝑏 𝑓 𝑠 𝑗 𝑘 𝑣 𝑥 𝑦 ℎ 𝑤 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mhphf.h	. . 3 ⊢ 𝐻 = (𝐼 mHomP 𝑈)
2		eqid 2759	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
3		eqid 2759	. . 3 ⊢ (0g‘𝑈) = (0g‘𝑈)
4		eqid 2759	. . 3 ⊢ (𝐼 mPoly 𝑈) = (𝐼 mPoly 𝑈)
5		eqid 2759	. . 3 ⊢ (+g‘(𝐼 mPoly 𝑈)) = (+g‘(𝐼 mPoly 𝑈))
6		eqid 2759	. . 3 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
7		eqid 2759	. . 3 ⊢ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} = {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}
8		mhphf.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
9		mhphf.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
10		mhphf.u	. . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅)
11	10	subrgring 19591	. . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
12	9, 11	syl 17	. . . 4 ⊢ (𝜑 → 𝑈 ∈ Ring)
13	12	ringgrpd 19359	. . 3 ⊢ (𝜑 → 𝑈 ∈ Grp)
14		mhphf.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
15		mhphf.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁))
16	4, 8, 12	mplsca 20761	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Scalar‘(𝐼 mPoly 𝑈)))
17	16	fveq2d 6655	. . . . . 6 ⊢ (𝜑 → (0g‘𝑈) = (0g‘(Scalar‘(𝐼 mPoly 𝑈))))
18	17	fveq2d 6655	. . . . 5 ⊢ (𝜑 → ((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)) = ((algSc‘(𝐼 mPoly 𝑈))‘(0g‘(Scalar‘(𝐼 mPoly 𝑈)))))
19		eqid 2759	. . . . . 6 ⊢ (algSc‘(𝐼 mPoly 𝑈)) = (algSc‘(𝐼 mPoly 𝑈))
20		eqid 2759	. . . . . 6 ⊢ (Scalar‘(𝐼 mPoly 𝑈)) = (Scalar‘(𝐼 mPoly 𝑈))
21	4	mpllmod 20767	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑈 ∈ Ring) → (𝐼 mPoly 𝑈) ∈ LMod)
22	8, 12, 21	syl2anc 588	. . . . . 6 ⊢ (𝜑 → (𝐼 mPoly 𝑈) ∈ LMod)
23	4	mplring 20768	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑈 ∈ Ring) → (𝐼 mPoly 𝑈) ∈ Ring)
24	8, 12, 23	syl2anc 588	. . . . . 6 ⊢ (𝜑 → (𝐼 mPoly 𝑈) ∈ Ring)
25	19, 20, 22, 24	ascl0 20631	. . . . 5 ⊢ (𝜑 → ((algSc‘(𝐼 mPoly 𝑈))‘(0g‘(Scalar‘(𝐼 mPoly 𝑈)))) = (0g‘(𝐼 mPoly 𝑈)))
26		eqid 2759	. . . . . 6 ⊢ (0g‘(𝐼 mPoly 𝑈)) = (0g‘(𝐼 mPoly 𝑈))
27	4, 6, 3, 26, 8, 13	mpl0 20756	. . . . 5 ⊢ (𝜑 → (0g‘(𝐼 mPoly 𝑈)) = ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝑈)}))
28	18, 25, 27	3eqtrrd 2799	. . . 4 ⊢ (𝜑 → ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝑈)}) = ((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)))
29		mhphf.m	. . . . . . . . . 10 ⊢ · = (.r‘𝑆)
30	10, 29	ressmulr 16668	. . . . . . . . 9 ⊢ (𝑅 ∈ (SubRing‘𝑆) → · = (.r‘𝑈))
31	9, 30	syl 17	. . . . . . . 8 ⊢ (𝜑 → · = (.r‘𝑈))
32	31	oveqd 7160	. . . . . . 7 ⊢ (𝜑 → ((𝑁 ↑ 𝐿) · (0g‘𝑈)) = ((𝑁 ↑ 𝐿)(.r‘𝑈)(0g‘𝑈)))
33		eqid 2759	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
34	33	subrgsubm 19601	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ∈ (SubMnd‘(mulGrp‘𝑆)))
35	9, 34	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (SubMnd‘(mulGrp‘𝑆)))
36		mhphf.l	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ 𝑅)
37		mhphf.e	. . . . . . . . . . 11 ⊢ ↑ = (.g‘(mulGrp‘𝑆))
38	37	submmulgcl 18322	. . . . . . . . . 10 ⊢ ((𝑅 ∈ (SubMnd‘(mulGrp‘𝑆)) ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ 𝑅) → (𝑁 ↑ 𝐿) ∈ 𝑅)
39	35, 14, 36, 38	syl3anc 1369	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 ↑ 𝐿) ∈ 𝑅)
40	10	subrgbas 19597	. . . . . . . . . 10 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 = (Base‘𝑈))
41	9, 40	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 = (Base‘𝑈))
42	39, 41	eleqtrd 2853	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ↑ 𝐿) ∈ (Base‘𝑈))
43		eqid 2759	. . . . . . . . 9 ⊢ (.r‘𝑈) = (.r‘𝑈)
44	2, 43, 3	ringrz 19394	. . . . . . . 8 ⊢ ((𝑈 ∈ Ring ∧ (𝑁 ↑ 𝐿) ∈ (Base‘𝑈)) → ((𝑁 ↑ 𝐿)(.r‘𝑈)(0g‘𝑈)) = (0g‘𝑈))
45	12, 42, 44	syl2anc 588	. . . . . . 7 ⊢ (𝜑 → ((𝑁 ↑ 𝐿)(.r‘𝑈)(0g‘𝑈)) = (0g‘𝑈))
46	32, 45	eqtrd 2794	. . . . . 6 ⊢ (𝜑 → ((𝑁 ↑ 𝐿) · (0g‘𝑈)) = (0g‘𝑈))
47		mhphf.q	. . . . . . . . 9 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
48		mhphf.k	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝑆)
49		eqid 2759	. . . . . . . . 9 ⊢ (Base‘(𝐼 mPoly 𝑈)) = (Base‘(𝐼 mPoly 𝑈))
50		mhphf.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ CRing)
51	2, 3	ring0cl 19375	. . . . . . . . . . 11 ⊢ (𝑈 ∈ Ring → (0g‘𝑈) ∈ (Base‘𝑈))
52	12, 51	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝑈) ∈ (Base‘𝑈))
53	52, 41	eleqtrrd 2854	. . . . . . . . 9 ⊢ (𝜑 → (0g‘𝑈) ∈ 𝑅)
54		mhphf.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))
55	47, 4, 10, 48, 49, 19, 8, 50, 9, 53, 54	evlsscaval 39763	. . . . . . . 8 ⊢ (𝜑 → (((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)) ∈ (Base‘(𝐼 mPoly 𝑈)) ∧ ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)))‘𝐴) = (0g‘𝑈)))
56	55	simprd 500	. . . . . . 7 ⊢ (𝜑 → ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)))‘𝐴) = (0g‘𝑈))
57	56	oveq2d 7159	. . . . . 6 ⊢ (𝜑 → ((𝑁 ↑ 𝐿) · ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)))‘𝐴)) = ((𝑁 ↑ 𝐿) · (0g‘𝑈)))
58	48	fvexi 6665	. . . . . . . . . 10 ⊢ 𝐾 ∈ V
59	58	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ V)
60	50	crngringd 19363	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ Ring)
61	48, 29	ringcl 19367	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Ring ∧ 𝑗 ∈ 𝐾 ∧ 𝑘 ∈ 𝐾) → (𝑗 · 𝑘) ∈ 𝐾)
62	60, 61	syl3an1 1161	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐾 ∧ 𝑘 ∈ 𝐾) → (𝑗 · 𝑘) ∈ 𝐾)
63	62	3expb 1118	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐾 ∧ 𝑘 ∈ 𝐾)) → (𝑗 · 𝑘) ∈ 𝐾)
64	48	subrgss 19589	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐾)
65	9, 64	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ⊆ 𝐾)
66	65, 36	sseldd 3889	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ 𝐾)
67		fconst6g 6546	. . . . . . . . . . 11 ⊢ (𝐿 ∈ 𝐾 → (𝐼 × {𝐿}):𝐼⟶𝐾)
68	66, 67	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼 × {𝐿}):𝐼⟶𝐾)
69		elmapi 8431	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝐾 ↑m 𝐼) → 𝐴:𝐼⟶𝐾)
70	54, 69	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴:𝐼⟶𝐾)
71		inidm 4119	. . . . . . . . . 10 ⊢ (𝐼 ∩ 𝐼) = 𝐼
72	63, 68, 70, 8, 8, 71	off 7415	. . . . . . . . 9 ⊢ (𝜑 → ((𝐼 × {𝐿}) ∘f · 𝐴):𝐼⟶𝐾)
73	59, 8, 72	elmapdd 39707	. . . . . . . 8 ⊢ (𝜑 → ((𝐼 × {𝐿}) ∘f · 𝐴) ∈ (𝐾 ↑m 𝐼))
74	47, 4, 10, 48, 49, 19, 8, 50, 9, 53, 73	evlsscaval 39763	. . . . . . 7 ⊢ (𝜑 → (((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)) ∈ (Base‘(𝐼 mPoly 𝑈)) ∧ ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)))‘((𝐼 × {𝐿}) ∘f · 𝐴)) = (0g‘𝑈)))
75	74	simprd 500	. . . . . 6 ⊢ (𝜑 → ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)))‘((𝐼 × {𝐿}) ∘f · 𝐴)) = (0g‘𝑈))
76	46, 57, 75	3eqtr4rd 2805	. . . . 5 ⊢ (𝜑 → ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)))‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)))‘𝐴)))
77		fvex 6664	. . . . . 6 ⊢ ((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)) ∈ V
78		fveq2 6651	. . . . . . . 8 ⊢ (𝑓 = ((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)) → (𝑄‘𝑓) = (𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈))))
79	78	fveq1d 6653	. . . . . . 7 ⊢ (𝑓 = ((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)) → ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)))‘((𝐼 × {𝐿}) ∘f · 𝐴)))
80	78	fveq1d 6653	. . . . . . . 8 ⊢ (𝑓 = ((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)) → ((𝑄‘𝑓)‘𝐴) = ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)))‘𝐴))
81	80	oveq2d 7159	. . . . . . 7 ⊢ (𝑓 = ((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)) → ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)))‘𝐴)))
82	79, 81	eqeq12d 2775	. . . . . 6 ⊢ (𝑓 = ((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)) → (((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴)) ↔ ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)))‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)))‘𝐴))))
83	77, 82	elab 3586	. . . . 5 ⊢ (((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)) ∈ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))} ↔ ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)))‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)))‘𝐴)))
84	76, 83	sylibr 237	. . . 4 ⊢ (𝜑 → ((algSc‘(𝐼 mPoly 𝑈))‘(0g‘𝑈)) ∈ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))})
85	28, 84	eqeltrd 2851	. . 3 ⊢ (𝜑 → ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝑈)}) ∈ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))})
86	10	subrgcrng 19592	. . . . . . . . . 10 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑈 ∈ CRing)
87	50, 9, 86	syl2anc 588	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ CRing)
88	4	mplassa 20771	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑈 ∈ CRing) → (𝐼 mPoly 𝑈) ∈ AssAlg)
89	8, 87, 88	syl2anc 588	. . . . . . . 8 ⊢ (𝜑 → (𝐼 mPoly 𝑈) ∈ AssAlg)
90	89	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (𝐼 mPoly 𝑈) ∈ AssAlg)
91	16	fveq2d 6655	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑈) = (Base‘(Scalar‘(𝐼 mPoly 𝑈))))
92	91	eleq2d 2836	. . . . . . . . 9 ⊢ (𝜑 → (𝑏 ∈ (Base‘𝑈) ↔ 𝑏 ∈ (Base‘(Scalar‘(𝐼 mPoly 𝑈)))))
93	92	biimpa 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (Base‘𝑈)) → 𝑏 ∈ (Base‘(Scalar‘(𝐼 mPoly 𝑈))))
94	93	adantrl 716	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → 𝑏 ∈ (Base‘(Scalar‘(𝐼 mPoly 𝑈))))
95		fvexd 6666	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝑈) ∈ V)
96		ovex 7176	. . . . . . . . . . . . 13 ⊢ (ℕ0 ↑m 𝐼) ∈ V
97	96	rabex 5195	. . . . . . . . . . . 12 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V
98	97	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V)
99		eqid 2759	. . . . . . . . . . . . . . . 16 ⊢ (1r‘𝑈) = (1r‘𝑈)
100	2, 99	ringidcl 19374	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ Ring → (1r‘𝑈) ∈ (Base‘𝑈))
101	12, 100	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1r‘𝑈) ∈ (Base‘𝑈))
102	101, 52	ifcld 4459	. . . . . . . . . . . . 13 ⊢ (𝜑 → if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)) ∈ (Base‘𝑈))
103	102	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)) ∈ (Base‘𝑈))
104	103	fmpttd 6863	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑈))
105	95, 98, 104	elmapdd 39707	. . . . . . . . . 10 ⊢ (𝜑 → (𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))) ∈ ((Base‘𝑈) ↑m {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}))
106		eqid 2759	. . . . . . . . . . 11 ⊢ (𝐼 mPwSer 𝑈) = (𝐼 mPwSer 𝑈)
107		eqid 2759	. . . . . . . . . . 11 ⊢ (Base‘(𝐼 mPwSer 𝑈)) = (Base‘(𝐼 mPwSer 𝑈))
108	106, 2, 6, 107, 8	psrbas 20691	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑈)) = ((Base‘𝑈) ↑m {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}))
109	105, 108	eleqtrrd 2854	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))) ∈ (Base‘(𝐼 mPwSer 𝑈)))
110	97	mptex 6970	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))) ∈ V
111	110	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))) ∈ V)
112		fvexd 6666	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝑈) ∈ V)
113		funmpt 6366	. . . . . . . . . . 11 ⊢ Fun (𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))
114	113	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → Fun (𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))))
115		snfi 8607	. . . . . . . . . . . 12 ⊢ {𝑎} ∈ Fin
116	115	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑎} ∈ Fin)
117		eldifsnneq 4674	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ {𝑎}) → ¬ 𝑤 = 𝑎)
118	117	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ {𝑎})) → ¬ 𝑤 = 𝑎)
119	118	iffalsed 4424	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∖ {𝑎})) → if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)) = (0g‘𝑈))
120	119, 98	suppss2 7867	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))) supp (0g‘𝑈)) ⊆ {𝑎})
121	116, 120	ssfid 8755	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))) supp (0g‘𝑈)) ∈ Fin)
122	111, 112, 114, 121	isfsuppd 39708	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))) finSupp (0g‘𝑈))
123	4, 106, 107, 3, 49	mplelbas 20743	. . . . . . . . 9 ⊢ ((𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))) ∈ (Base‘(𝐼 mPoly 𝑈)) ↔ ((𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))) ∈ (Base‘(𝐼 mPwSer 𝑈)) ∧ (𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))) finSupp (0g‘𝑈)))
124	109, 122, 123	sylanbrc 587	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))) ∈ (Base‘(𝐼 mPoly 𝑈)))
125	124	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))) ∈ (Base‘(𝐼 mPoly 𝑈)))
126		eqid 2759	. . . . . . . 8 ⊢ (Base‘(Scalar‘(𝐼 mPoly 𝑈))) = (Base‘(Scalar‘(𝐼 mPoly 𝑈)))
127		eqid 2759	. . . . . . . 8 ⊢ (.r‘(𝐼 mPoly 𝑈)) = (.r‘(𝐼 mPoly 𝑈))
128		eqid 2759	. . . . . . . 8 ⊢ ( ·𝑠 ‘(𝐼 mPoly 𝑈)) = ( ·𝑠 ‘(𝐼 mPoly 𝑈))
129	19, 20, 126, 49, 127, 128	asclmul1 20633	. . . . . . 7 ⊢ (((𝐼 mPoly 𝑈) ∈ AssAlg ∧ 𝑏 ∈ (Base‘(Scalar‘(𝐼 mPoly 𝑈))) ∧ (𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))) ∈ (Base‘(𝐼 mPoly 𝑈))) → (((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))) = (𝑏( ·𝑠 ‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))))
130	90, 94, 125, 129	syl3anc 1369	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))) = (𝑏( ·𝑠 ‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))))
131		simprr 773	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → 𝑏 ∈ (Base‘𝑈))
132	4, 128, 2, 49, 43, 6, 131, 125	mplvsca 20763	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (𝑏( ·𝑠 ‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))) = (({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑏}) ∘f (.r‘𝑈)(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))))
133	130, 132	eqtrd 2794	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))) = (({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑏}) ∘f (.r‘𝑈)(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))))
134		vex 3411	. . . . . . 7 ⊢ 𝑏 ∈ V
135		fnconstg 6545	. . . . . . 7 ⊢ (𝑏 ∈ V → ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑏}) Fn {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
136	134, 135	mp1i 13	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑏}) Fn {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
137		fvex 6664	. . . . . . . . 9 ⊢ (1r‘𝑈) ∈ V
138		fvex 6664	. . . . . . . . 9 ⊢ (0g‘𝑈) ∈ V
139	137, 138	ifex 4463	. . . . . . . 8 ⊢ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)) ∈ V
140		eqid 2759	. . . . . . . 8 ⊢ (𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))) = (𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))
141	139, 140	fnmpti 6467	. . . . . . 7 ⊢ (𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))) Fn {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
142	141	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))) Fn {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
143	97	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V)
144		inidm 4119	. . . . . 6 ⊢ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∩ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
145	134	fvconst2 6950	. . . . . . 7 ⊢ (𝑠 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → (({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑏})‘𝑠) = 𝑏)
146	145	adantl 486	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑠 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑏})‘𝑠) = 𝑏)
147		equequ1 2033	. . . . . . . . 9 ⊢ (𝑤 = 𝑠 → (𝑤 = 𝑎 ↔ 𝑠 = 𝑎))
148	147	ifbid 4436	. . . . . . . 8 ⊢ (𝑤 = 𝑠 → if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)) = if(𝑠 = 𝑎, (1r‘𝑈), (0g‘𝑈)))
149	137, 138	ifex 4463	. . . . . . . 8 ⊢ if(𝑠 = 𝑎, (1r‘𝑈), (0g‘𝑈)) ∈ V
150	148, 140, 149	fvmpt 6752	. . . . . . 7 ⊢ (𝑠 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → ((𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))‘𝑠) = if(𝑠 = 𝑎, (1r‘𝑈), (0g‘𝑈)))
151	150	adantl 486	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑠 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → ((𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))‘𝑠) = if(𝑠 = 𝑎, (1r‘𝑈), (0g‘𝑈)))
152	136, 142, 143, 143, 144, 146, 151	offval 7406	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑏}) ∘f (.r‘𝑈)(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))) = (𝑠 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏(.r‘𝑈)if(𝑠 = 𝑎, (1r‘𝑈), (0g‘𝑈)))))
153		ovif2 7239	. . . . . . 7 ⊢ (𝑏(.r‘𝑈)if(𝑠 = 𝑎, (1r‘𝑈), (0g‘𝑈))) = if(𝑠 = 𝑎, (𝑏(.r‘𝑈)(1r‘𝑈)), (𝑏(.r‘𝑈)(0g‘𝑈)))
154	12	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑠 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑈 ∈ Ring)
155		simplrr 778	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑠 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑏 ∈ (Base‘𝑈))
156	2, 43, 99	ringridm 19378	. . . . . . . . 9 ⊢ ((𝑈 ∈ Ring ∧ 𝑏 ∈ (Base‘𝑈)) → (𝑏(.r‘𝑈)(1r‘𝑈)) = 𝑏)
157	154, 155, 156	syl2anc 588	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑠 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑏(.r‘𝑈)(1r‘𝑈)) = 𝑏)
158	2, 43, 3	ringrz 19394	. . . . . . . . 9 ⊢ ((𝑈 ∈ Ring ∧ 𝑏 ∈ (Base‘𝑈)) → (𝑏(.r‘𝑈)(0g‘𝑈)) = (0g‘𝑈))
159	154, 155, 158	syl2anc 588	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑠 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑏(.r‘𝑈)(0g‘𝑈)) = (0g‘𝑈))
160	157, 159	ifeq12d 4434	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑠 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → if(𝑠 = 𝑎, (𝑏(.r‘𝑈)(1r‘𝑈)), (𝑏(.r‘𝑈)(0g‘𝑈))) = if(𝑠 = 𝑎, 𝑏, (0g‘𝑈)))
161	153, 160	syl5eq 2806	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑠 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑏(.r‘𝑈)if(𝑠 = 𝑎, (1r‘𝑈), (0g‘𝑈))) = if(𝑠 = 𝑎, 𝑏, (0g‘𝑈)))
162	161	mpteq2dva 5120	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (𝑠 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑏(.r‘𝑈)if(𝑠 = 𝑎, (1r‘𝑈), (0g‘𝑈)))) = (𝑠 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑠 = 𝑎, 𝑏, (0g‘𝑈))))
163	133, 152, 162	3eqtrd 2798	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))) = (𝑠 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑠 = 𝑎, 𝑏, (0g‘𝑈))))
164	8	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → 𝐼 ∈ 𝑉)
165	50	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → 𝑆 ∈ CRing)
166	9	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → 𝑅 ∈ (SubRing‘𝑆))
167	73	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((𝐼 × {𝐿}) ∘f · 𝐴) ∈ (𝐾 ↑m 𝐼))
168	12	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → 𝑈 ∈ Ring)
169	4, 49, 2, 19, 164, 168	mplasclf 20811	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (algSc‘(𝐼 mPoly 𝑈)):(Base‘𝑈)⟶(Base‘(𝐼 mPoly 𝑈)))
170	169, 131	ffvelrnd 6836	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((algSc‘(𝐼 mPoly 𝑈))‘𝑏) ∈ (Base‘(𝐼 mPoly 𝑈)))
171		eqidd 2760	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘𝑏))‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘𝑏))‘((𝐼 × {𝐿}) ∘f · 𝐴)))
172	170, 171	jca 516	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (((algSc‘(𝐼 mPoly 𝑈))‘𝑏) ∈ (Base‘(𝐼 mPoly 𝑈)) ∧ ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘𝑏))‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘𝑏))‘((𝐼 × {𝐿}) ∘f · 𝐴))))
173		elrabi 3594	. . . . . . . . . 10 ⊢ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} → 𝑎 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
174	173	ad2antrl 728	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → 𝑎 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
175	47, 4, 10, 49, 48, 33, 37, 3, 99, 6, 140, 164, 165, 166, 167, 174	evlsbagval 39765	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))) ∈ (Base‘(𝐼 mPoly 𝑈)) ∧ ((𝑄‘(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))))‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (((𝐼 × {𝐿}) ∘f · 𝐴)‘𝑣))))))
176	47, 4, 10, 48, 49, 164, 165, 166, 167, 172, 175, 127, 29	evlsmulval 39768	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))) ∈ (Base‘(𝐼 mPoly 𝑈)) ∧ ((𝑄‘(((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))))‘((𝐼 × {𝐿}) ∘f · 𝐴)) = (((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘𝑏))‘((𝐼 × {𝐿}) ∘f · 𝐴)) · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (((𝐼 × {𝐿}) ∘f · 𝐴)‘𝑣)))))))
177	176	simprd 500	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((𝑄‘(((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))))‘((𝐼 × {𝐿}) ∘f · 𝐴)) = (((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘𝑏))‘((𝐼 × {𝐿}) ∘f · 𝐴)) · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (((𝐼 × {𝐿}) ∘f · 𝐴)‘𝑣))))))
178	33	ringmgp 19356	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd)
179	60, 178	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ Mnd)
180	33, 48	mgpbas 19298	. . . . . . . . . . . . 13 ⊢ 𝐾 = (Base‘(mulGrp‘𝑆))
181	180, 37	mulgnn0cl 18296	. . . . . . . . . . . 12 ⊢ (((mulGrp‘𝑆) ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝐿 ∈ 𝐾) → (𝑁 ↑ 𝐿) ∈ 𝐾)
182	179, 14, 66, 181	syl3anc 1369	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 ↑ 𝐿) ∈ 𝐾)
183	182	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (𝑁 ↑ 𝐿) ∈ 𝐾)
184	41, 65	eqsstrrd 3927	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝑈) ⊆ 𝐾)
185	184	sselda 3888	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ (Base‘𝑈)) → 𝑏 ∈ 𝐾)
186	185	adantrl 716	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → 𝑏 ∈ 𝐾)
187	48, 29	crngcom 19368	. . . . . . . . . 10 ⊢ ((𝑆 ∈ CRing ∧ (𝑁 ↑ 𝐿) ∈ 𝐾 ∧ 𝑏 ∈ 𝐾) → ((𝑁 ↑ 𝐿) · 𝑏) = (𝑏 · (𝑁 ↑ 𝐿)))
188	165, 183, 186, 187	syl3anc 1369	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((𝑁 ↑ 𝐿) · 𝑏) = (𝑏 · (𝑁 ↑ 𝐿)))
189	188	oveq1d 7158	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (((𝑁 ↑ 𝐿) · 𝑏) · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))) = ((𝑏 · (𝑁 ↑ 𝐿)) · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))))
190	165	crngringd 19363	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → 𝑆 ∈ Ring)
191		eqid 2759	. . . . . . . . . . 11 ⊢ (1r‘𝑆) = (1r‘𝑆)
192	33, 191	ringidval 19306	. . . . . . . . . 10 ⊢ (1r‘𝑆) = (0g‘(mulGrp‘𝑆))
193	33	crngmgp 19358	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ CRing → (mulGrp‘𝑆) ∈ CMnd)
194	50, 193	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ CMnd)
195	194	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (mulGrp‘𝑆) ∈ CMnd)
196	179	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑣 ∈ 𝐼) → (mulGrp‘𝑆) ∈ Mnd)
197		elrabi 3594	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑎 ∈ (ℕ0 ↑m 𝐼))
198		elmapi 8431	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ (ℕ0 ↑m 𝐼) → 𝑎:𝐼⟶ℕ0)
199	173, 197, 198	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} → 𝑎:𝐼⟶ℕ0)
200	199	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → 𝑎:𝐼⟶ℕ0)
201	200	adantrr 717	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → 𝑎:𝐼⟶ℕ0)
202	201	ffvelrnda 6835	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑣 ∈ 𝐼) → (𝑎‘𝑣) ∈ ℕ0)
203	70	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑣 ∈ 𝐼) → 𝐴:𝐼⟶𝐾)
204		simpr 489	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑣 ∈ 𝐼) → 𝑣 ∈ 𝐼)
205	203, 204	ffvelrnd 6836	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑣 ∈ 𝐼) → (𝐴‘𝑣) ∈ 𝐾)
206	180, 37	mulgnn0cl 18296	. . . . . . . . . . . 12 ⊢ (((mulGrp‘𝑆) ∈ Mnd ∧ (𝑎‘𝑣) ∈ ℕ0 ∧ (𝐴‘𝑣) ∈ 𝐾) → ((𝑎‘𝑣) ↑ (𝐴‘𝑣)) ∈ 𝐾)
207	196, 202, 205, 206	syl3anc 1369	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑣 ∈ 𝐼) → ((𝑎‘𝑣) ↑ (𝐴‘𝑣)) ∈ 𝐾)
208	207	fmpttd 6863	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))):𝐼⟶𝐾)
209	6	psrbagfsupp 20667	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} → 𝑎 finSupp 0)
210	209	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → 𝑎 finSupp 0)
211	210	fsuppimpd 8858	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}) → (𝑎 supp 0) ∈ Fin)
212	173, 211	sylan2 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → (𝑎 supp 0) ∈ Fin)
213	212	adantrr 717	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (𝑎 supp 0) ∈ Fin)
214	201	feqmptd 6714	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → 𝑎 = (𝑣 ∈ 𝐼 ↦ (𝑎‘𝑣)))
215	214	oveq1d 7158	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (𝑎 supp 0) = ((𝑣 ∈ 𝐼 ↦ (𝑎‘𝑣)) supp 0))
216		ssidd 3911	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (𝑎 supp 0) ⊆ (𝑎 supp 0))
217	215, 216	eqsstrrd 3927	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((𝑣 ∈ 𝐼 ↦ (𝑎‘𝑣)) supp 0) ⊆ (𝑎 supp 0))
218	180, 192, 37	mulg0 18283	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝐾 → (0 ↑ 𝑘) = (1r‘𝑆))
219	218	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑘 ∈ 𝐾) → (0 ↑ 𝑘) = (1r‘𝑆))
220		c0ex 10658	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
221	220	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → 0 ∈ V)
222	217, 219, 202, 205, 221	suppssov1 7865	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))) supp (1r‘𝑆)) ⊆ (𝑎 supp 0))
223	213, 222	ssfid 8755	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))) supp (1r‘𝑆)) ∈ Fin)
224	180, 192, 195, 164, 208, 223	gsumcl2 19087	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣)))) ∈ 𝐾)
225	48, 29	ringass 19370	. . . . . . . . 9 ⊢ ((𝑆 ∈ Ring ∧ ((𝑁 ↑ 𝐿) ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣)))) ∈ 𝐾)) → (((𝑁 ↑ 𝐿) · 𝑏) · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))) = ((𝑁 ↑ 𝐿) · (𝑏 · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣)))))))
226	190, 183, 186, 224, 225	syl13anc 1370	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (((𝑁 ↑ 𝐿) · 𝑏) · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))) = ((𝑁 ↑ 𝐿) · (𝑏 · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣)))))))
227	48, 29	ringass 19370	. . . . . . . . 9 ⊢ ((𝑆 ∈ Ring ∧ (𝑏 ∈ 𝐾 ∧ (𝑁 ↑ 𝐿) ∈ 𝐾 ∧ ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣)))) ∈ 𝐾)) → ((𝑏 · (𝑁 ↑ 𝐿)) · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))) = (𝑏 · ((𝑁 ↑ 𝐿) · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣)))))))
228	190, 186, 183, 224, 227	syl13anc 1370	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((𝑏 · (𝑁 ↑ 𝐿)) · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))) = (𝑏 · ((𝑁 ↑ 𝐿) · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣)))))))
229	189, 226, 228	3eqtr3d 2802	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((𝑁 ↑ 𝐿) · (𝑏 · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣)))))) = (𝑏 · ((𝑁 ↑ 𝐿) · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣)))))))
230	54	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → 𝐴 ∈ (𝐾 ↑m 𝐼))
231	41	eleq2d 2836	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑏 ∈ 𝑅 ↔ 𝑏 ∈ (Base‘𝑈)))
232	231	biimpar 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (Base‘𝑈)) → 𝑏 ∈ 𝑅)
233	232	adantrl 716	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → 𝑏 ∈ 𝑅)
234	47, 4, 10, 48, 49, 19, 164, 165, 166, 233, 230	evlsscaval 39763	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (((algSc‘(𝐼 mPoly 𝑈))‘𝑏) ∈ (Base‘(𝐼 mPoly 𝑈)) ∧ ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘𝑏))‘𝐴) = 𝑏))
235	47, 4, 10, 49, 48, 33, 37, 3, 99, 6, 140, 164, 165, 166, 230, 174	evlsbagval 39765	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))) ∈ (Base‘(𝐼 mPoly 𝑈)) ∧ ((𝑄‘(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))))‘𝐴) = ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))))
236	47, 4, 10, 48, 49, 164, 165, 166, 230, 234, 235, 127, 29	evlsmulval 39768	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))) ∈ (Base‘(𝐼 mPoly 𝑈)) ∧ ((𝑄‘(((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))))‘𝐴) = (𝑏 · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣)))))))
237	236	simprd 500	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((𝑄‘(((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))))‘𝐴) = (𝑏 · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))))
238	237	oveq2d 7159	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((𝑁 ↑ 𝐿) · ((𝑄‘(((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))))‘𝐴)) = ((𝑁 ↑ 𝐿) · (𝑏 · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣)))))))
239	47, 4, 10, 48, 49, 19, 164, 165, 166, 233, 167	evlsscaval 39763	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (((algSc‘(𝐼 mPoly 𝑈))‘𝑏) ∈ (Base‘(𝐼 mPoly 𝑈)) ∧ ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘𝑏))‘((𝐼 × {𝐿}) ∘f · 𝐴)) = 𝑏))
240	239	simprd 500	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘𝑏))‘((𝐼 × {𝐿}) ∘f · 𝐴)) = 𝑏)
241		fconst6g 6546	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 ∈ 𝑅 → (𝐼 × {𝐿}):𝐼⟶𝑅)
242	36, 241	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐼 × {𝐿}):𝐼⟶𝑅)
243	242	ffnd 6492	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐼 × {𝐿}) Fn 𝐼)
244	243	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (𝐼 × {𝐿}) Fn 𝐼)
245	70	ffnd 6492	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 Fn 𝐼)
246	245	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → 𝐴 Fn 𝐼)
247	36	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑣 ∈ 𝐼) → 𝐿 ∈ 𝑅)
248		fvconst2g 6948	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 ∈ 𝑅 ∧ 𝑣 ∈ 𝐼) → ((𝐼 × {𝐿})‘𝑣) = 𝐿)
249	247, 204, 248	syl2anc 588	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑣 ∈ 𝐼) → ((𝐼 × {𝐿})‘𝑣) = 𝐿)
250		eqidd 2760	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑣 ∈ 𝐼) → (𝐴‘𝑣) = (𝐴‘𝑣))
251	244, 246, 164, 164, 71, 249, 250	ofval 7408	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑣 ∈ 𝐼) → (((𝐼 × {𝐿}) ∘f · 𝐴)‘𝑣) = (𝐿 · (𝐴‘𝑣)))
252	251	oveq2d 7159	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑣 ∈ 𝐼) → ((𝑎‘𝑣) ↑ (((𝐼 × {𝐿}) ∘f · 𝐴)‘𝑣)) = ((𝑎‘𝑣) ↑ (𝐿 · (𝐴‘𝑣))))
253	194	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑣 ∈ 𝐼) → (mulGrp‘𝑆) ∈ CMnd)
254	66	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑣 ∈ 𝐼) → 𝐿 ∈ 𝐾)
255	33, 29	mgpplusg 19296	. . . . . . . . . . . . . 14 ⊢ · = (+g‘(mulGrp‘𝑆))
256	180, 37, 255	mulgnn0di 18999	. . . . . . . . . . . . 13 ⊢ (((mulGrp‘𝑆) ∈ CMnd ∧ ((𝑎‘𝑣) ∈ ℕ0 ∧ 𝐿 ∈ 𝐾 ∧ (𝐴‘𝑣) ∈ 𝐾)) → ((𝑎‘𝑣) ↑ (𝐿 · (𝐴‘𝑣))) = (((𝑎‘𝑣) ↑ 𝐿) · ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))
257	253, 202, 254, 205, 256	syl13anc 1370	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑣 ∈ 𝐼) → ((𝑎‘𝑣) ↑ (𝐿 · (𝐴‘𝑣))) = (((𝑎‘𝑣) ↑ 𝐿) · ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))
258	252, 257	eqtrd 2794	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) ∧ 𝑣 ∈ 𝐼) → ((𝑎‘𝑣) ↑ (((𝐼 × {𝐿}) ∘f · 𝐴)‘𝑣)) = (((𝑎‘𝑣) ↑ 𝐿) · ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))
259	258	mpteq2dva 5120	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (((𝐼 × {𝐿}) ∘f · 𝐴)‘𝑣))) = (𝑣 ∈ 𝐼 ↦ (((𝑎‘𝑣) ↑ 𝐿) · ((𝑎‘𝑣) ↑ (𝐴‘𝑣)))))
260	259	oveq2d 7159	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (((𝐼 × {𝐿}) ∘f · 𝐴)‘𝑣)))) = ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ (((𝑎‘𝑣) ↑ 𝐿) · ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))))
261	194	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → (mulGrp‘𝑆) ∈ CMnd)
262	8	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → 𝐼 ∈ 𝑉)
263	179	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) ∧ 𝑣 ∈ 𝐼) → (mulGrp‘𝑆) ∈ Mnd)
264	200	ffvelrnda 6835	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) ∧ 𝑣 ∈ 𝐼) → (𝑎‘𝑣) ∈ ℕ0)
265	66	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) ∧ 𝑣 ∈ 𝐼) → 𝐿 ∈ 𝐾)
266	180, 37	mulgnn0cl 18296	. . . . . . . . . . . . 13 ⊢ (((mulGrp‘𝑆) ∈ Mnd ∧ (𝑎‘𝑣) ∈ ℕ0 ∧ 𝐿 ∈ 𝐾) → ((𝑎‘𝑣) ↑ 𝐿) ∈ 𝐾)
267	263, 264, 265, 266	syl3anc 1369	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) ∧ 𝑣 ∈ 𝐼) → ((𝑎‘𝑣) ↑ 𝐿) ∈ 𝐾)
268	70	ffvelrnda 6835	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → (𝐴‘𝑣) ∈ 𝐾)
269	268	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) ∧ 𝑣 ∈ 𝐼) → (𝐴‘𝑣) ∈ 𝐾)
270	263, 264, 269, 206	syl3anc 1369	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) ∧ 𝑣 ∈ 𝐼) → ((𝑎‘𝑣) ↑ (𝐴‘𝑣)) ∈ 𝐾)
271		eqidd 2760	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ 𝐿)) = (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ 𝐿)))
272		eqidd 2760	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))) = (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))
273	262	mptexd 6971	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ 𝐿)) ∈ V)
274		fvexd 6666	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → (1r‘𝑆) ∈ V)
275		funmpt 6366	. . . . . . . . . . . . . 14 ⊢ Fun (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ 𝐿))
276	275	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → Fun (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ 𝐿)))
277	200	feqmptd 6714	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → 𝑎 = (𝑣 ∈ 𝐼 ↦ (𝑎‘𝑣)))
278	277	oveq1d 7158	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → (𝑎 supp 0) = ((𝑣 ∈ 𝐼 ↦ (𝑎‘𝑣)) supp 0))
279		ssidd 3911	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → (𝑎 supp 0) ⊆ (𝑎 supp 0))
280	278, 279	eqsstrrd 3927	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → ((𝑣 ∈ 𝐼 ↦ (𝑎‘𝑣)) supp 0) ⊆ (𝑎 supp 0))
281	218	adantl 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) ∧ 𝑘 ∈ 𝐾) → (0 ↑ 𝑘) = (1r‘𝑆))
282	220	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → 0 ∈ V)
283	280, 281, 264, 265, 282	suppssov1 7865	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → ((𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ 𝐿)) supp (1r‘𝑆)) ⊆ (𝑎 supp 0))
284	212, 283	ssfid 8755	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → ((𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ 𝐿)) supp (1r‘𝑆)) ∈ Fin)
285	273, 274, 276, 284	isfsuppd 39708	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ 𝐿)) finSupp (1r‘𝑆))
286	262	mptexd 6971	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))) ∈ V)
287		funmpt 6366	. . . . . . . . . . . . . 14 ⊢ Fun (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣)))
288	287	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → Fun (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))
289	280, 281, 264, 269, 282	suppssov1 7865	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → ((𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))) supp (1r‘𝑆)) ⊆ (𝑎 supp 0))
290	212, 289	ssfid 8755	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → ((𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))) supp (1r‘𝑆)) ∈ Fin)
291	286, 274, 288, 290	isfsuppd 39708	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))) finSupp (1r‘𝑆))
292	180, 192, 255, 261, 262, 267, 270, 271, 272, 285, 291	gsummptfsadd 19097	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ (((𝑎‘𝑣) ↑ 𝐿) · ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))) = (((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ 𝐿))) · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))))
293	6, 7, 180, 37, 8, 179, 66, 14	mhphflem 39774	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ 𝐿))) = (𝑁 ↑ 𝐿))
294	293	oveq1d 7158	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → (((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ 𝐿))) · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))) = ((𝑁 ↑ 𝐿) · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))))
295	292, 294	eqtrd 2794	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) → ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ (((𝑎‘𝑣) ↑ 𝐿) · ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))) = ((𝑁 ↑ 𝐿) · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))))
296	295	adantrr 717	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ (((𝑎‘𝑣) ↑ 𝐿) · ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))) = ((𝑁 ↑ 𝐿) · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))))
297	260, 296	eqtrd 2794	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (((𝐼 × {𝐿}) ∘f · 𝐴)‘𝑣)))) = ((𝑁 ↑ 𝐿) · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣))))))
298	240, 297	oveq12d 7161	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘𝑏))‘((𝐼 × {𝐿}) ∘f · 𝐴)) · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (((𝐼 × {𝐿}) ∘f · 𝐴)‘𝑣))))) = (𝑏 · ((𝑁 ↑ 𝐿) · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (𝐴‘𝑣)))))))
299	229, 238, 298	3eqtr4rd 2805	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (((𝑄‘((algSc‘(𝐼 mPoly 𝑈))‘𝑏))‘((𝐼 × {𝐿}) ∘f · 𝐴)) · ((mulGrp‘𝑆) Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) ↑ (((𝐼 × {𝐿}) ∘f · 𝐴)‘𝑣))))) = ((𝑁 ↑ 𝐿) · ((𝑄‘(((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))))‘𝐴)))
300	177, 299	eqtrd 2794	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → ((𝑄‘(((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))))‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘(((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))))‘𝐴)))
301		ovex 7176	. . . . . 6 ⊢ (((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))) ∈ V
302		fveq2 6651	. . . . . . . 8 ⊢ (𝑓 = (((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))) → (𝑄‘𝑓) = (𝑄‘(((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈))))))
303	302	fveq1d 6653	. . . . . . 7 ⊢ (𝑓 = (((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))) → ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑄‘(((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))))‘((𝐼 × {𝐿}) ∘f · 𝐴)))
304	302	fveq1d 6653	. . . . . . . 8 ⊢ (𝑓 = (((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))) → ((𝑄‘𝑓)‘𝐴) = ((𝑄‘(((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))))‘𝐴))
305	304	oveq2d 7159	. . . . . . 7 ⊢ (𝑓 = (((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))) → ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘(((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))))‘𝐴)))
306	303, 305	eqeq12d 2775	. . . . . 6 ⊢ (𝑓 = (((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))) → (((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴)) ↔ ((𝑄‘(((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))))‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘(((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))))‘𝐴))))
307	301, 306	elab 3586	. . . . 5 ⊢ ((((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))) ∈ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))} ↔ ((𝑄‘(((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))))‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘(((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))))‘𝐴)))
308	300, 307	sylibr 237	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (((algSc‘(𝐼 mPoly 𝑈))‘𝑏)(.r‘(𝐼 mPoly 𝑈))(𝑤 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑤 = 𝑎, (1r‘𝑈), (0g‘𝑈)))) ∈ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))})
309	163, 308	eqeltrrd 2852	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁} ∧ 𝑏 ∈ (Base‘𝑈))) → (𝑠 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑠 = 𝑎, 𝑏, (0g‘𝑈))) ∈ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))})
310		elin 3870	. . . . . 6 ⊢ (𝑥 ∈ ((𝐻‘𝑁) ∩ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))}) ↔ (𝑥 ∈ (𝐻‘𝑁) ∧ 𝑥 ∈ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))}))
311		vex 3411	. . . . . . . 8 ⊢ 𝑥 ∈ V
312		fveq2 6651	. . . . . . . . . 10 ⊢ (𝑓 = 𝑥 → (𝑄‘𝑓) = (𝑄‘𝑥))
313	312	fveq1d 6653	. . . . . . . . 9 ⊢ (𝑓 = 𝑥 → ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)))
314	312	fveq1d 6653	. . . . . . . . . 10 ⊢ (𝑓 = 𝑥 → ((𝑄‘𝑓)‘𝐴) = ((𝑄‘𝑥)‘𝐴))
315	314	oveq2d 7159	. . . . . . . . 9 ⊢ (𝑓 = 𝑥 → ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴)))
316	313, 315	eqeq12d 2775	. . . . . . . 8 ⊢ (𝑓 = 𝑥 → (((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴)) ↔ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))))
317	311, 316	elab 3586	. . . . . . 7 ⊢ (𝑥 ∈ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))} ↔ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴)))
318	317	anbi2i 626	. . . . . 6 ⊢ ((𝑥 ∈ (𝐻‘𝑁) ∧ 𝑥 ∈ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))}) ↔ (𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))))
319	310, 318	bitri 278	. . . . 5 ⊢ (𝑥 ∈ ((𝐻‘𝑁) ∩ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))}) ↔ (𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))))
320		elin 3870	. . . . . 6 ⊢ (𝑦 ∈ ((𝐻‘𝑁) ∩ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))}) ↔ (𝑦 ∈ (𝐻‘𝑁) ∧ 𝑦 ∈ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))}))
321		vex 3411	. . . . . . . 8 ⊢ 𝑦 ∈ V
322		fveq2 6651	. . . . . . . . . 10 ⊢ (𝑓 = 𝑦 → (𝑄‘𝑓) = (𝑄‘𝑦))
323	322	fveq1d 6653	. . . . . . . . 9 ⊢ (𝑓 = 𝑦 → ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)))
324	322	fveq1d 6653	. . . . . . . . . 10 ⊢ (𝑓 = 𝑦 → ((𝑄‘𝑓)‘𝐴) = ((𝑄‘𝑦)‘𝐴))
325	324	oveq2d 7159	. . . . . . . . 9 ⊢ (𝑓 = 𝑦 → ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴)))
326	323, 325	eqeq12d 2775	. . . . . . . 8 ⊢ (𝑓 = 𝑦 → (((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴)) ↔ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))
327	321, 326	elab 3586	. . . . . . 7 ⊢ (𝑦 ∈ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))} ↔ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴)))
328	327	anbi2i 626	. . . . . 6 ⊢ ((𝑦 ∈ (𝐻‘𝑁) ∧ 𝑦 ∈ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))}) ↔ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))
329	320, 328	bitri 278	. . . . 5 ⊢ (𝑦 ∈ ((𝐻‘𝑁) ∩ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))}) ↔ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))
330	319, 329	anbi12i 630	. . . 4 ⊢ ((𝑥 ∈ ((𝐻‘𝑁) ∩ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))}) ∧ 𝑦 ∈ ((𝐻‘𝑁) ∩ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))})) ↔ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴)))))
331	60	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → 𝑆 ∈ Ring)
332	182	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → (𝑁 ↑ 𝐿) ∈ 𝐾)
333		eqid 2759	. . . . . . . . 9 ⊢ (𝑆 ↑s (Base‘(𝑆 ↑s 𝐼))) = (𝑆 ↑s (Base‘(𝑆 ↑s 𝐼)))
334		eqid 2759	. . . . . . . . 9 ⊢ (Base‘(𝑆 ↑s (Base‘(𝑆 ↑s 𝐼)))) = (Base‘(𝑆 ↑s (Base‘(𝑆 ↑s 𝐼))))
335	50	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → 𝑆 ∈ CRing)
336		fvexd 6666	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → (Base‘(𝑆 ↑s 𝐼)) ∈ V)
337		eqid 2759	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ↑s (𝐾 ↑m 𝐼)) = (𝑆 ↑s (𝐾 ↑m 𝐼))
338	47, 4, 10, 337, 48	evlsrhm 20836	. . . . . . . . . . . . . 14 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ ((𝐼 mPoly 𝑈) RingHom (𝑆 ↑s (𝐾 ↑m 𝐼))))
339	8, 50, 9, 338	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 ∈ ((𝐼 mPoly 𝑈) RingHom (𝑆 ↑s (𝐾 ↑m 𝐼))))
340		eqid 2759	. . . . . . . . . . . . . 14 ⊢ (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))) = (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼)))
341	49, 340	rhmf 19534	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ ((𝐼 mPoly 𝑈) RingHom (𝑆 ↑s (𝐾 ↑m 𝐼))) → 𝑄:(Base‘(𝐼 mPoly 𝑈))⟶(Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))))
342	339, 341	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄:(Base‘(𝐼 mPoly 𝑈))⟶(Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))))
343	342	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → 𝑄:(Base‘(𝐼 mPoly 𝑈))⟶(Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))))
344	8	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝐼 ∈ 𝑉)
345	12	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝑈 ∈ Ring)
346	14	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝑁 ∈ ℕ0)
347		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝑥 ∈ (𝐻‘𝑁))
348	1, 4, 49, 344, 345, 346, 347	mhpmpl 20872	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐻‘𝑁)) → 𝑥 ∈ (Base‘(𝐼 mPoly 𝑈)))
349	348	adantrr 717	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴)))) → 𝑥 ∈ (Base‘(𝐼 mPoly 𝑈)))
350	349	adantrr 717	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → 𝑥 ∈ (Base‘(𝐼 mPoly 𝑈)))
351	343, 350	ffvelrnd 6836	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → (𝑄‘𝑥) ∈ (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))))
352		eqid 2759	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ↑s 𝐼) = (𝑆 ↑s 𝐼)
353	352, 48	pwsbas 16803	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ CRing ∧ 𝐼 ∈ 𝑉) → (𝐾 ↑m 𝐼) = (Base‘(𝑆 ↑s 𝐼)))
354	50, 8, 353	syl2anc 588	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐾 ↑m 𝐼) = (Base‘(𝑆 ↑s 𝐼)))
355	354	oveq2d 7159	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 ↑s (𝐾 ↑m 𝐼)) = (𝑆 ↑s (Base‘(𝑆 ↑s 𝐼))))
356	355	fveq2d 6655	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))) = (Base‘(𝑆 ↑s (Base‘(𝑆 ↑s 𝐼)))))
357	356	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))) = (Base‘(𝑆 ↑s (Base‘(𝑆 ↑s 𝐼)))))
358	351, 357	eleqtrd 2853	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → (𝑄‘𝑥) ∈ (Base‘(𝑆 ↑s (Base‘(𝑆 ↑s 𝐼)))))
359	333, 48, 334, 335, 336, 358	pwselbas 16805	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → (𝑄‘𝑥):(Base‘(𝑆 ↑s 𝐼))⟶𝐾)
360	54, 354	eleqtrd 2853	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (Base‘(𝑆 ↑s 𝐼)))
361	360	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → 𝐴 ∈ (Base‘(𝑆 ↑s 𝐼)))
362	359, 361	ffvelrnd 6836	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → ((𝑄‘𝑥)‘𝐴) ∈ 𝐾)
363	8	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝐼 ∈ 𝑉)
364	12	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝑈 ∈ Ring)
365	14	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝑁 ∈ ℕ0)
366		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝑦 ∈ (𝐻‘𝑁))
367	1, 4, 49, 363, 364, 365, 366	mhpmpl 20872	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐻‘𝑁)) → 𝑦 ∈ (Base‘(𝐼 mPoly 𝑈)))
368	367	adantrr 717	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴)))) → 𝑦 ∈ (Base‘(𝐼 mPoly 𝑈)))
369	368	adantrl 716	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → 𝑦 ∈ (Base‘(𝐼 mPoly 𝑈)))
370	343, 369	ffvelrnd 6836	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → (𝑄‘𝑦) ∈ (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))))
371	370, 357	eleqtrd 2853	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → (𝑄‘𝑦) ∈ (Base‘(𝑆 ↑s (Base‘(𝑆 ↑s 𝐼)))))
372	333, 48, 334, 335, 336, 371	pwselbas 16805	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → (𝑄‘𝑦):(Base‘(𝑆 ↑s 𝐼))⟶𝐾)
373	372, 361	ffvelrnd 6836	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → ((𝑄‘𝑦)‘𝐴) ∈ 𝐾)
374		eqid 2759	. . . . . . . 8 ⊢ (+g‘𝑆) = (+g‘𝑆)
375	48, 374, 29	ringdi 19372	. . . . . . 7 ⊢ ((𝑆 ∈ Ring ∧ ((𝑁 ↑ 𝐿) ∈ 𝐾 ∧ ((𝑄‘𝑥)‘𝐴) ∈ 𝐾 ∧ ((𝑄‘𝑦)‘𝐴) ∈ 𝐾)) → ((𝑁 ↑ 𝐿) · (((𝑄‘𝑥)‘𝐴)(+g‘𝑆)((𝑄‘𝑦)‘𝐴))) = (((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))(+g‘𝑆)((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))
376	331, 332, 362, 373, 375	syl13anc 1370	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → ((𝑁 ↑ 𝐿) · (((𝑄‘𝑥)‘𝐴)(+g‘𝑆)((𝑄‘𝑦)‘𝐴))) = (((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))(+g‘𝑆)((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))
377	8	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → 𝐼 ∈ 𝑉)
378	9	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → 𝑅 ∈ (SubRing‘𝑆))
379	54	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → 𝐴 ∈ (𝐾 ↑m 𝐼))
380		eqidd 2760	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) → ((𝑄‘𝑥)‘𝐴) = ((𝑄‘𝑥)‘𝐴))
381	348, 380	anim12dan 622	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴)))) → (𝑥 ∈ (Base‘(𝐼 mPoly 𝑈)) ∧ ((𝑄‘𝑥)‘𝐴) = ((𝑄‘𝑥)‘𝐴)))
382	381	adantrr 717	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → (𝑥 ∈ (Base‘(𝐼 mPoly 𝑈)) ∧ ((𝑄‘𝑥)‘𝐴) = ((𝑄‘𝑥)‘𝐴)))
383		eqidd 2760	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))) → ((𝑄‘𝑦)‘𝐴) = ((𝑄‘𝑦)‘𝐴))
384	367, 383	anim12dan 622	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴)))) → (𝑦 ∈ (Base‘(𝐼 mPoly 𝑈)) ∧ ((𝑄‘𝑦)‘𝐴) = ((𝑄‘𝑦)‘𝐴)))
385	384	adantrl 716	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → (𝑦 ∈ (Base‘(𝐼 mPoly 𝑈)) ∧ ((𝑄‘𝑦)‘𝐴) = ((𝑄‘𝑦)‘𝐴)))
386	47, 4, 10, 48, 49, 377, 335, 378, 379, 382, 385, 5, 374	evlsaddval 39767	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → ((𝑥(+g‘(𝐼 mPoly 𝑈))𝑦) ∈ (Base‘(𝐼 mPoly 𝑈)) ∧ ((𝑄‘(𝑥(+g‘(𝐼 mPoly 𝑈))𝑦))‘𝐴) = (((𝑄‘𝑥)‘𝐴)(+g‘𝑆)((𝑄‘𝑦)‘𝐴))))
387	386	simprd 500	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → ((𝑄‘(𝑥(+g‘(𝐼 mPoly 𝑈))𝑦))‘𝐴) = (((𝑄‘𝑥)‘𝐴)(+g‘𝑆)((𝑄‘𝑦)‘𝐴)))
388	387	oveq2d 7159	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → ((𝑁 ↑ 𝐿) · ((𝑄‘(𝑥(+g‘(𝐼 mPoly 𝑈))𝑦))‘𝐴)) = ((𝑁 ↑ 𝐿) · (((𝑄‘𝑥)‘𝐴)(+g‘𝑆)((𝑄‘𝑦)‘𝐴))))
389	58	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → 𝐾 ∈ V)
390	63	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) ∧ (𝑗 ∈ 𝐾 ∧ 𝑘 ∈ 𝐾)) → (𝑗 · 𝑘) ∈ 𝐾)
391	68	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → (𝐼 × {𝐿}):𝐼⟶𝐾)
392	70	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → 𝐴:𝐼⟶𝐾)
393	390, 391, 392, 377, 377, 71	off 7415	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → ((𝐼 × {𝐿}) ∘f · 𝐴):𝐼⟶𝐾)
394	389, 377, 393	elmapdd 39707	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → ((𝐼 × {𝐿}) ∘f · 𝐴) ∈ (𝐾 ↑m 𝐼))
395		simpr 489	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) → ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴)))
396	348, 395	anim12dan 622	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴)))) → (𝑥 ∈ (Base‘(𝐼 mPoly 𝑈)) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))))
397	396	adantrr 717	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → (𝑥 ∈ (Base‘(𝐼 mPoly 𝑈)) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))))
398		simpr 489	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))) → ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴)))
399	367, 398	anim12dan 622	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴)))) → (𝑦 ∈ (Base‘(𝐼 mPoly 𝑈)) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))
400	399	adantrl 716	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → (𝑦 ∈ (Base‘(𝐼 mPoly 𝑈)) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))
401	47, 4, 10, 48, 49, 377, 335, 378, 394, 397, 400, 5, 374	evlsaddval 39767	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → ((𝑥(+g‘(𝐼 mPoly 𝑈))𝑦) ∈ (Base‘(𝐼 mPoly 𝑈)) ∧ ((𝑄‘(𝑥(+g‘(𝐼 mPoly 𝑈))𝑦))‘((𝐼 × {𝐿}) ∘f · 𝐴)) = (((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))(+g‘𝑆)((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴)))))
402	401	simprd 500	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → ((𝑄‘(𝑥(+g‘(𝐼 mPoly 𝑈))𝑦))‘((𝐼 × {𝐿}) ∘f · 𝐴)) = (((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))(+g‘𝑆)((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))
403	376, 388, 402	3eqtr4rd 2805	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → ((𝑄‘(𝑥(+g‘(𝐼 mPoly 𝑈))𝑦))‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘(𝑥(+g‘(𝐼 mPoly 𝑈))𝑦))‘𝐴)))
404		ovex 7176	. . . . . 6 ⊢ (𝑥(+g‘(𝐼 mPoly 𝑈))𝑦) ∈ V
405		fveq2 6651	. . . . . . . 8 ⊢ (𝑓 = (𝑥(+g‘(𝐼 mPoly 𝑈))𝑦) → (𝑄‘𝑓) = (𝑄‘(𝑥(+g‘(𝐼 mPoly 𝑈))𝑦)))
406	405	fveq1d 6653	. . . . . . 7 ⊢ (𝑓 = (𝑥(+g‘(𝐼 mPoly 𝑈))𝑦) → ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑄‘(𝑥(+g‘(𝐼 mPoly 𝑈))𝑦))‘((𝐼 × {𝐿}) ∘f · 𝐴)))
407	405	fveq1d 6653	. . . . . . . 8 ⊢ (𝑓 = (𝑥(+g‘(𝐼 mPoly 𝑈))𝑦) → ((𝑄‘𝑓)‘𝐴) = ((𝑄‘(𝑥(+g‘(𝐼 mPoly 𝑈))𝑦))‘𝐴))
408	407	oveq2d 7159	. . . . . . 7 ⊢ (𝑓 = (𝑥(+g‘(𝐼 mPoly 𝑈))𝑦) → ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘(𝑥(+g‘(𝐼 mPoly 𝑈))𝑦))‘𝐴)))
409	406, 408	eqeq12d 2775	. . . . . 6 ⊢ (𝑓 = (𝑥(+g‘(𝐼 mPoly 𝑈))𝑦) → (((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴)) ↔ ((𝑄‘(𝑥(+g‘(𝐼 mPoly 𝑈))𝑦))‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘(𝑥(+g‘(𝐼 mPoly 𝑈))𝑦))‘𝐴))))
410	404, 409	elab 3586	. . . . 5 ⊢ ((𝑥(+g‘(𝐼 mPoly 𝑈))𝑦) ∈ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))} ↔ ((𝑄‘(𝑥(+g‘(𝐼 mPoly 𝑈))𝑦))‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘(𝑥(+g‘(𝐼 mPoly 𝑈))𝑦))‘𝐴)))
411	403, 410	sylibr 237	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑥)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑥)‘𝐴))) ∧ (𝑦 ∈ (𝐻‘𝑁) ∧ ((𝑄‘𝑦)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑦)‘𝐴))))) → (𝑥(+g‘(𝐼 mPoly 𝑈))𝑦) ∈ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))})
412	330, 411	sylan2b 597	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐻‘𝑁) ∩ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))}) ∧ 𝑦 ∈ ((𝐻‘𝑁) ∩ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))}))) → (𝑥(+g‘(𝐼 mPoly 𝑈))𝑦) ∈ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))})
413	1, 2, 3, 4, 5, 6, 7, 8, 13, 14, 15, 85, 309, 412	mhpind 39773	. 2 ⊢ (𝜑 → 𝑋 ∈ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))})
414		fveq2 6651	. . . . . 6 ⊢ (𝑓 = 𝑋 → (𝑄‘𝑓) = (𝑄‘𝑋))
415	414	fveq1d 6653	. . . . 5 ⊢ (𝑓 = 𝑋 → ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘f · 𝐴)))
416	414	fveq1d 6653	. . . . . 6 ⊢ (𝑓 = 𝑋 → ((𝑄‘𝑓)‘𝐴) = ((𝑄‘𝑋)‘𝐴))
417	416	oveq2d 7159	. . . . 5 ⊢ (𝑓 = 𝑋 → ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
418	415, 417	eqeq12d 2775	. . . 4 ⊢ (𝑓 = 𝑋 → (((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴)) ↔ ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴))))
419	418	elabg 3585	. . 3 ⊢ (𝑋 ∈ (𝐻‘𝑁) → (𝑋 ∈ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))} ↔ ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴))))
420	15, 419	syl 17	. 2 ⊢ (𝜑 → (𝑋 ∈ {𝑓 ∣ ((𝑄‘𝑓)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑓)‘𝐴))} ↔ ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴))))
421	413, 420	mpbid 235	1 ⊢ (𝜑 → ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  {cab 2736  {crab 3072  Vcvv 3407   ∖ cdif 3851   ∩ cin 3853   ⊆ wss 3854  ifcif 4413  {csn 4515   class class class wbr 5025   ↦ cmpt 5105   × cxp 5515  ^◡ccnv 5516   “ cima 5520  Fun wfun 6322   Fn wfn 6323  ⟶wf 6324  ‘cfv 6328  (class class class)co 7143   ∘_f cof 7396   supp csupp 7828   ↑_m cmap 8409  Fincfn 8520   finSupp cfsupp 8851  0cc0 10560  ℕcn 11659  ℕ₀cn0 11919  Basecbs 16526   ↾_s cress 16527  +_gcplusg 16608  ._rcmulr 16609  Scalarcsca 16611   ·_𝑠 cvsca 16612  0_gc0g 16756   Σ_g cgsu 16757   ↑_s cpws 16763  Mndcmnd 17962  SubMndcsubmnd 18006  ._gcmg 18276  CMndccmn 18958  mulGrpcmgp 19292  1_rcur 19304  Ringcrg 19350  CRingccrg 19351   RingHom crh 19520  SubRingcsubrg 19584  LModclmod 19687  ℂ_fldccnfld 20151  AssAlgcasa 20600  algSccascl 20602   mPwSer cmps 20651   mPoly cmpl 20653   evalSub ces 20818   mHomP cmhp 20857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5149  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452  ax-cnex 10616  ax-resscn 10617  ax-1cn 10618  ax-icn 10619  ax-addcl 10620  ax-addrcl 10621  ax-mulcl 10622  ax-mulrcl 10623  ax-mulcom 10624  ax-addass 10625  ax-mulass 10626  ax-distr 10627  ax-i2m1 10628  ax-1ne0 10629  ax-1rid 10630  ax-rnegex 10631  ax-rrecex 10632  ax-cnre 10633  ax-pre-lttri 10634  ax-pre-lttrn 10635  ax-pre-ltadd 10636  ax-pre-mulgt0 10637  ax-addf 10639  ax-mulf 10640

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-nel 3054  df-ral 3073  df-rex 3074  df-reu 3075  df-rmo 3076  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-pss 3873  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-tp 4520  df-op 4522  df-uni 4792  df-int 4832  df-iun 4878  df-iin 4879  df-br 5026  df-opab 5088  df-mpt 5106  df-tr 5132  df-id 5423  df-eprel 5428  df-po 5436  df-so 5437  df-fr 5476  df-se 5477  df-we 5478  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7101  df-ov 7146  df-oprab 7147  df-mpo 7148  df-of 7398  df-ofr 7399  df-om 7573  df-1st 7686  df-2nd 7687  df-supp 7829  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-2o 8106  df-oadd 8109  df-er 8292  df-map 8411  df-pm 8412  df-ixp 8473  df-en 8521  df-dom 8522  df-sdom 8523  df-fin 8524  df-fsupp 8852  df-sup 8924  df-oi 8992  df-dju 9348  df-card 9386  df-pnf 10700  df-mnf 10701  df-xr 10702  df-ltxr 10703  df-le 10704  df-sub 10895  df-neg 10896  df-nn 11660  df-2 11722  df-3 11723  df-4 11724  df-5 11725  df-6 11726  df-7 11727  df-8 11728  df-9 11729  df-n0 11920  df-z 12006  df-dec 12123  df-uz 12268  df-fz 12925  df-fzo 13068  df-seq 13404  df-hash 13726  df-struct 16528  df-ndx 16529  df-slot 16530  df-base 16532  df-sets 16533  df-ress 16534  df-plusg 16621  df-mulr 16622  df-starv 16623  df-sca 16624  df-vsca 16625  df-ip 16626  df-tset 16627  df-ple 16628  df-ds 16630  df-unif 16631  df-hom 16632  df-cco 16633  df-0g 16758  df-gsum 16759  df-prds 16764  df-pws 16766  df-mre 16900  df-mrc 16901  df-acs 16903  df-mgm 17903  df-sgrp 17952  df-mnd 17963  df-mhm 18007  df-submnd 18008  df-grp 18157  df-minusg 18158  df-sbg 18159  df-mulg 18277  df-subg 18328  df-ghm 18408  df-cntz 18499  df-cmn 18960  df-abl 18961  df-mgp 19293  df-ur 19305  df-srg 19309  df-ring 19352  df-cring 19353  df-rnghom 19523  df-subrg 19586  df-lmod 19689  df-lss 19757  df-lsp 19797  df-cnfld 20152  df-assa 20603  df-asp 20604  df-ascl 20605  df-psr 20656  df-mvr 20657  df-mpl 20658  df-evls 20820  df-mhp 20861

This theorem is referenced by: (None)