evlsvvval - Mathbox for Steven Nguyen

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Theorem evlsvvval 41861

Description: Give a formula for the evaluation of a polynomial given assignments from variables to values. This is the sum of the evaluations for each term (corresponding to a bag of variables), that is, the coefficient times the product of each variable raised to the corresponding power. (Contributed by SN, 5-Mar-2025.)

**Hypotheses**
Ref	Expression
evlsvvval.q	⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
evlsvvval.p	⊢ 𝑃 = (𝐼 mPoly 𝑈)
evlsvvval.b	⊢ 𝐵 = (Base‘𝑃)
evlsvvval.u	⊢ 𝑈 = (𝑆 ↾_s 𝑅)
evlsvvval.d	⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
evlsvvval.k	⊢ 𝐾 = (Base‘𝑆)
evlsvvval.m	⊢ 𝑀 = (mulGrp‘𝑆)
evlsvvval.w	⊢ ↑ = (._g‘𝑀)
evlsvvval.x	⊢ · = (._r‘𝑆)
evlsvvval.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
evlsvvval.s	⊢ (𝜑 → 𝑆 ∈ CRing)
evlsvvval.r	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
evlsvvval.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
evlsvvval.a	⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))

**Assertion**
Ref	Expression
evlsvvval	⊢ (𝜑 → ((𝑄‘𝐹)‘𝐴) = (𝑆 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))))

Distinct variable groups: 𝜑,𝑏,𝑖 𝑃,𝑏 𝐵,𝑏 𝐷,𝑏,𝑖 𝐾,𝑏,𝑖 𝑈,𝑏,ℎ 𝐹,𝑏 𝑖,𝐼,𝑏 ℎ,𝐼 𝑆,𝑏,𝑖 𝑅,𝑏 𝐴,𝑏,𝑖 Allowed substitution hints: 𝜑(ℎ) 𝐴(ℎ) 𝐵(ℎ,𝑖) 𝐷(ℎ) 𝑃(ℎ,𝑖) 𝑄(ℎ,𝑖,𝑏) 𝑅(ℎ,𝑖) 𝑆(ℎ) · (ℎ,𝑖,𝑏) 𝑈(𝑖) ↑ (ℎ,𝑖,𝑏) 𝐹(ℎ,𝑖) 𝐾(ℎ) 𝑀(ℎ,𝑖,𝑏) 𝑉(ℎ,𝑖,𝑏)

Proof of Theorem evlsvvval Dummy variables 𝑎 𝑥 𝑝 𝑙 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6891	. . . . . . . 8 ⊢ (𝑙 = 𝐴 → (𝑙‘𝑖) = (𝐴‘𝑖))
2	1	oveq2d 7432	. . . . . . 7 ⊢ (𝑙 = 𝐴 → ((𝑏‘𝑖) ↑ (𝑙‘𝑖)) = ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))
3	2	mpteq2dv 5245	. . . . . 6 ⊢ (𝑙 = 𝐴 → (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖))) = (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))
4	3	oveq2d 7432	. . . . 5 ⊢ (𝑙 = 𝐴 → (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖)))) = (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))
5	4	oveq2d 7432	. . . 4 ⊢ (𝑙 = 𝐴 → ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖))))) = ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))
6	5	mpteq2dv 5245	. . 3 ⊢ (𝑙 = 𝐴 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖)))))) = (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))))
7	6	oveq2d 7432	. 2 ⊢ (𝑙 = 𝐴 → (𝑆 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖))))))) = (𝑆 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))))
8		evlsvvval.q	. . . 4 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
9		evlsvvval.p	. . . 4 ⊢ 𝑃 = (𝐼 mPoly 𝑈)
10		evlsvvval.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
11		evlsvvval.d	. . . 4 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
12		evlsvvval.k	. . . 4 ⊢ 𝐾 = (Base‘𝑆)
13		evlsvvval.u	. . . 4 ⊢ 𝑈 = (𝑆 ↾_s 𝑅)
14		eqid 2725	. . . 4 ⊢ (𝑆 ↑_s (𝐾 ↑_m 𝐼)) = (𝑆 ↑_s (𝐾 ↑_m 𝐼))
15		eqid 2725	. . . 4 ⊢ (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
16		eqid 2725	. . . 4 ⊢ (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))) = (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
17		eqid 2725	. . . 4 ⊢ (._r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (._r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
18		eqid 2725	. . . 4 ⊢ (𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥})) = (𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))
19		eqid 2725	. . . 4 ⊢ (𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)))
20		evlsvvval.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
21		evlsvvval.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ CRing)
22		evlsvvval.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
23		evlsvvval.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
24	8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23	evlsvval 41858	. . 3 ⊢ (𝜑 → (𝑄‘𝐹) = ((𝑆 ↑_s (𝐾 ↑_m 𝐼)) Σ_g (𝑏 ∈ 𝐷 ↦ (((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝑏))(._r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))((mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) Σ_g (𝑏 ∘_f (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)))))))))
25		sneq 4634	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑏) → {𝑥} = {(𝐹‘𝑏)})
26	25	xpeq2d 5702	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑏) → ((𝐾 ↑_m 𝐼) × {𝑥}) = ((𝐾 ↑_m 𝐼) × {(𝐹‘𝑏)}))
27		eqid 2725	. . . . . . . . . . 11 ⊢ (Base‘𝑈) = (Base‘𝑈)
28	9, 27, 10, 11, 23	mplelf 21947	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐷⟶(Base‘𝑈))
29	13	subrgbas 20524	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 = (Base‘𝑈))
30	22, 29	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 = (Base‘𝑈))
31	30	feq3d 6704	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹:𝐷⟶𝑅 ↔ 𝐹:𝐷⟶(Base‘𝑈)))
32	28, 31	mpbird 256	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐷⟶𝑅)
33	32	ffvelcdmda 7089	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘𝑏) ∈ 𝑅)
34		ovex 7449	. . . . . . . . . 10 ⊢ (𝐾 ↑_m 𝐼) ∈ V
35		snex 5427	. . . . . . . . . 10 ⊢ {(𝐹‘𝑏)} ∈ V
36	34, 35	xpex 7753	. . . . . . . . 9 ⊢ ((𝐾 ↑_m 𝐼) × {(𝐹‘𝑏)}) ∈ V
37	36	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((𝐾 ↑_m 𝐼) × {(𝐹‘𝑏)}) ∈ V)
38	18, 26, 33, 37	fvmptd3 7023	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝑏)) = ((𝐾 ↑_m 𝐼) × {(𝐹‘𝑏)}))
39	11	psrbagf 21855	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ 𝐷 → 𝑏:𝐼⟶ℕ₀)
40	39	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑏:𝐼⟶ℕ₀)
41	40	ffnd 6718	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑏 Fn 𝐼)
42	34	mptex 7231	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)) ∈ V
43	42, 19	fnmpti 6693	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥))) Fn 𝐼
44	43	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥))) Fn 𝐼)
45	20	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐼 ∈ 𝑉)
46		inidm 4213	. . . . . . . . . . 11 ⊢ (𝐼 ∩ 𝐼) = 𝐼
47		eqidd 2726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) → (𝑏‘𝑖) = (𝑏‘𝑖))
48		fveq2 6892	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑖 → (𝑎‘𝑥) = (𝑎‘𝑖))
49	48	mpteq2dv 5245	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑖 → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)) = (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑖)))
50		simpr 483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) → 𝑖 ∈ 𝐼)
51		eqid 2725	. . . . . . . . . . . . 13 ⊢ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
52	21	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ CRing)
53		ovexd 7451	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) → (𝐾 ↑_m 𝐼) ∈ V)
54		elmapi 8866	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ (𝐾 ↑_m 𝐼) → 𝑎:𝐼⟶𝐾)
55	54	ffvelcdmda 7089	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑖 ∈ 𝐼) → (𝑎‘𝑖) ∈ 𝐾)
56	55	ancoms 457	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ 𝐼 ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → (𝑎‘𝑖) ∈ 𝐾)
57	56	adantll 712	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → (𝑎‘𝑖) ∈ 𝐾)
58	57	fmpttd 7120	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑖)):(𝐾 ↑_m 𝐼)⟶𝐾)
59	14, 12, 51, 52, 53, 58	pwselbasr 41829	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑖)) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
60	19, 49, 50, 59	fvmptd3 7023	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) → ((𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)))‘𝑖) = (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑖)))
61	41, 44, 45, 45, 46, 47, 60	offval 7691	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑏 ∘_f (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)))) = (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖)(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑖)))))
62	15, 51	mgpbas 20084	. . . . . . . . . . . . . . 15 ⊢ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (Base‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
63	21	crngringd 20190	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑆 ∈ Ring)
64		ovexd 7451	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐾 ↑_m 𝐼) ∈ V)
65	14	pwsring 20264	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∈ Ring ∧ (𝐾 ↑_m 𝐼) ∈ V) → (𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ Ring)
66	63, 64, 65	syl2anc 582	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ Ring)
67	15	ringmgp 20183	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ Ring → (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∈ Mnd)
68	66, 67	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∈ Mnd)
69	68	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) → (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∈ Mnd)
70	40	ffvelcdmda 7089	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) → (𝑏‘𝑖) ∈ ℕ₀)
71	62, 16, 69, 70, 59	mulgnn0cld 19054	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) → ((𝑏‘𝑖)(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑖))) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
72	14, 12, 51, 52, 53, 71	pwselbas 17470	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) → ((𝑏‘𝑖)(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑖))):(𝐾 ↑_m 𝐼)⟶𝐾)
73	72	ffnd 6718	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) → ((𝑏‘𝑖)(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑖))) Fn (𝐾 ↑_m 𝐼))
74		ovex 7449	. . . . . . . . . . . . . 14 ⊢ ((𝑏‘𝑖) ↑ (𝑚‘𝑖)) ∈ V
75		eqid 2725	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖))) = (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖)))
76	74, 75	fnmpti 6693	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖))) Fn (𝐾 ↑_m 𝐼)
77	76	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) → (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖))) Fn (𝐾 ↑_m 𝐼))
78		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑖)) = (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑖))
79		fveq1 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑝 → (𝑎‘𝑖) = (𝑝‘𝑖))
80		simpr 483	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) ∧ 𝑝 ∈ (𝐾 ↑_m 𝐼)) → 𝑝 ∈ (𝐾 ↑_m 𝐼))
81		fvexd 6907	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) ∧ 𝑝 ∈ (𝐾 ↑_m 𝐼)) → (𝑝‘𝑖) ∈ V)
82	78, 79, 80, 81	fvmptd3 7023	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) ∧ 𝑝 ∈ (𝐾 ↑_m 𝐼)) → ((𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑖))‘𝑝) = (𝑝‘𝑖))
83	82	oveq2d 7432	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) ∧ 𝑝 ∈ (𝐾 ↑_m 𝐼)) → ((𝑏‘𝑖) ↑ ((𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑖))‘𝑝)) = ((𝑏‘𝑖) ↑ (𝑝‘𝑖)))
84		evlsvvval.m	. . . . . . . . . . . . . 14 ⊢ 𝑀 = (mulGrp‘𝑆)
85		evlsvvval.w	. . . . . . . . . . . . . 14 ⊢ ↑ = (._g‘𝑀)
86	63	ad3antrrr 728	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) ∧ 𝑝 ∈ (𝐾 ↑_m 𝐼)) → 𝑆 ∈ Ring)
87		ovexd 7451	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) ∧ 𝑝 ∈ (𝐾 ↑_m 𝐼)) → (𝐾 ↑_m 𝐼) ∈ V)
88	70	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) ∧ 𝑝 ∈ (𝐾 ↑_m 𝐼)) → (𝑏‘𝑖) ∈ ℕ₀)
89	59	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) ∧ 𝑝 ∈ (𝐾 ↑_m 𝐼)) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑖)) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
90	14, 51, 15, 84, 16, 85, 86, 87, 88, 89, 80	pwsexpg 20269	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) ∧ 𝑝 ∈ (𝐾 ↑_m 𝐼)) → (((𝑏‘𝑖)(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑖)))‘𝑝) = ((𝑏‘𝑖) ↑ ((𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑖))‘𝑝)))
91		fveq1 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑝 → (𝑚‘𝑖) = (𝑝‘𝑖))
92	91	oveq2d 7432	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑝 → ((𝑏‘𝑖) ↑ (𝑚‘𝑖)) = ((𝑏‘𝑖) ↑ (𝑝‘𝑖)))
93		ovexd 7451	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) ∧ 𝑝 ∈ (𝐾 ↑_m 𝐼)) → ((𝑏‘𝑖) ↑ (𝑝‘𝑖)) ∈ V)
94	75, 92, 80, 93	fvmptd3 7023	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) ∧ 𝑝 ∈ (𝐾 ↑_m 𝐼)) → ((𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖)))‘𝑝) = ((𝑏‘𝑖) ↑ (𝑝‘𝑖)))
95	83, 90, 94	3eqtr4d 2775	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) ∧ 𝑝 ∈ (𝐾 ↑_m 𝐼)) → (((𝑏‘𝑖)(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑖)))‘𝑝) = ((𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖)))‘𝑝))
96	73, 77, 95	eqfnfvd 7038	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ 𝐼) → ((𝑏‘𝑖)(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑖))) = (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖))))
97	96	mpteq2dva 5243	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖)(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑖)))) = (𝑖 ∈ 𝐼 ↦ (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖)))))
98	61, 97	eqtrd 2765	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑏 ∘_f (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)))) = (𝑖 ∈ 𝐼 ↦ (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖)))))
99	98	oveq2d 7432	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) Σ_g (𝑏 ∘_f (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥))))) = ((mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) Σ_g (𝑖 ∈ 𝐼 ↦ (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖))))))
100		eqid 2725	. . . . . . . . 9 ⊢ (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
101		ovexd 7451	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐾 ↑_m 𝐼) ∈ V)
102	21	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑆 ∈ CRing)
103	84, 12	mgpbas 20084	. . . . . . . . . 10 ⊢ 𝐾 = (Base‘𝑀)
104	84	ringmgp 20183	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Ring → 𝑀 ∈ Mnd)
105	63, 104	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ Mnd)
106	105	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ (𝑚 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑖 ∈ 𝐼)) → 𝑀 ∈ Mnd)
107	70	adantrl 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ (𝑚 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑖 ∈ 𝐼)) → (𝑏‘𝑖) ∈ ℕ₀)
108		elmapi 8866	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (𝐾 ↑_m 𝐼) → 𝑚:𝐼⟶𝐾)
109	108	ffvelcdmda 7089	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑖 ∈ 𝐼) → (𝑚‘𝑖) ∈ 𝐾)
110	109	adantl 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ (𝑚 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑖 ∈ 𝐼)) → (𝑚‘𝑖) ∈ 𝐾)
111	103, 85, 106, 107, 110	mulgnn0cld 19054	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ (𝑚 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑖 ∈ 𝐼)) → ((𝑏‘𝑖) ↑ (𝑚‘𝑖)) ∈ 𝐾)
112	45	mptexd 7232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑖 ∈ 𝐼 ↦ (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖)))) ∈ V)
113		fvexd 6907	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∈ V)
114		funmpt 6586	. . . . . . . . . . 11 ⊢ Fun (𝑖 ∈ 𝐼 ↦ (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖))))
115	114	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → Fun (𝑖 ∈ 𝐼 ↦ (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖)))))
116	11	psrbagfsupp 21857	. . . . . . . . . . 11 ⊢ (𝑏 ∈ 𝐷 → 𝑏 finSupp 0)
117	116	adantl 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑏 finSupp 0)
118		ssidd 3996	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑏 supp 0) ⊆ (𝑏 supp 0))
119		0cnd 11237	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 0 ∈ ℂ)
120	40, 118, 45, 119	suppssr 8199	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ (𝐼 ∖ (𝑏 supp 0))) → (𝑏‘𝑖) = 0)
121	120	oveq1d 7431	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ (𝐼 ∖ (𝑏 supp 0))) → ((𝑏‘𝑖) ↑ (𝑚‘𝑖)) = (0 ↑ (𝑚‘𝑖)))
122	121	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ (𝐼 ∖ (𝑏 supp 0))) ∧ 𝑚 ∈ (𝐾 ↑_m 𝐼)) → ((𝑏‘𝑖) ↑ (𝑚‘𝑖)) = (0 ↑ (𝑚‘𝑖)))
123		eldifi 4119	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (𝐼 ∖ (𝑏 supp 0)) → 𝑖 ∈ 𝐼)
124	123, 109	sylan2 591	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑖 ∈ (𝐼 ∖ (𝑏 supp 0))) → (𝑚‘𝑖) ∈ 𝐾)
125	124	ancoms 457	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (𝐼 ∖ (𝑏 supp 0)) ∧ 𝑚 ∈ (𝐾 ↑_m 𝐼)) → (𝑚‘𝑖) ∈ 𝐾)
126	125	adantll 712	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ (𝐼 ∖ (𝑏 supp 0))) ∧ 𝑚 ∈ (𝐾 ↑_m 𝐼)) → (𝑚‘𝑖) ∈ 𝐾)
127		eqid 2725	. . . . . . . . . . . . . . . . 17 ⊢ (1_r‘𝑆) = (1_r‘𝑆)
128	84, 127	ringidval 20127	. . . . . . . . . . . . . . . 16 ⊢ (1_r‘𝑆) = (0_g‘𝑀)
129	103, 128, 85	mulg0 19034	. . . . . . . . . . . . . . 15 ⊢ ((𝑚‘𝑖) ∈ 𝐾 → (0 ↑ (𝑚‘𝑖)) = (1_r‘𝑆))
130	126, 129	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ (𝐼 ∖ (𝑏 supp 0))) ∧ 𝑚 ∈ (𝐾 ↑_m 𝐼)) → (0 ↑ (𝑚‘𝑖)) = (1_r‘𝑆))
131	122, 130	eqtrd 2765	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ (𝐼 ∖ (𝑏 supp 0))) ∧ 𝑚 ∈ (𝐾 ↑_m 𝐼)) → ((𝑏‘𝑖) ↑ (𝑚‘𝑖)) = (1_r‘𝑆))
132	131	mpteq2dva 5243	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ (𝐼 ∖ (𝑏 supp 0))) → (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖))) = (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ (1_r‘𝑆)))
133		fconstmpt 5734	. . . . . . . . . . . . 13 ⊢ ((𝐾 ↑_m 𝐼) × {(1_r‘𝑆)}) = (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ (1_r‘𝑆))
134	14, 127	pws1 20265	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Ring ∧ (𝐾 ↑_m 𝐼) ∈ V) → ((𝐾 ↑_m 𝐼) × {(1_r‘𝑆)}) = (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
135	63, 64, 134	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐾 ↑_m 𝐼) × {(1_r‘𝑆)}) = (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
136	135	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ (𝐼 ∖ (𝑏 supp 0))) → ((𝐾 ↑_m 𝐼) × {(1_r‘𝑆)}) = (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
137	133, 136	eqtr3id 2779	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ (𝐼 ∖ (𝑏 supp 0))) → (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ (1_r‘𝑆)) = (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
138	132, 137	eqtrd 2765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑖 ∈ (𝐼 ∖ (𝑏 supp 0))) → (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖))) = (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
139	138, 45	suppss2 8204	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((𝑖 ∈ 𝐼 ↦ (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖)))) supp (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))) ⊆ (𝑏 supp 0))
140	112, 113, 115, 117, 139	fsuppsssuppgd 9405	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑖 ∈ 𝐼 ↦ (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖)))) finSupp (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
141	14, 12, 100, 15, 84, 101, 45, 102, 111, 140	pwsgprod 41830	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) Σ_g (𝑖 ∈ 𝐼 ↦ (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖))))) = (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖))))))
142	99, 141	eqtrd 2765	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) Σ_g (𝑏 ∘_f (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥))))) = (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖))))))
143	38, 142	oveq12d 7434	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝑏))(._r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))((mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) Σ_g (𝑏 ∘_f (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)))))) = (((𝐾 ↑_m 𝐼) × {(𝐹‘𝑏)})(._r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))(𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖)))))))
144	12	subrgss 20515	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐾)
145	29, 144	eqsstrrd 4012	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (Base‘𝑈) ⊆ 𝐾)
146	22, 145	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝑈) ⊆ 𝐾)
147	28, 146	fssd 6735	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐷⟶𝐾)
148	147	ffvelcdmda 7089	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘𝑏) ∈ 𝐾)
149		fconst6g 6781	. . . . . . . . 9 ⊢ ((𝐹‘𝑏) ∈ 𝐾 → ((𝐾 ↑_m 𝐼) × {(𝐹‘𝑏)}):(𝐾 ↑_m 𝐼)⟶𝐾)
150	148, 149	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((𝐾 ↑_m 𝐼) × {(𝐹‘𝑏)}):(𝐾 ↑_m 𝐼)⟶𝐾)
151	14, 12, 51, 102, 101, 150	pwselbasr 41829	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((𝐾 ↑_m 𝐼) × {(𝐹‘𝑏)}) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
152	20	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑚 ∈ (𝐾 ↑_m 𝐼)) → 𝐼 ∈ 𝑉)
153	21	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑚 ∈ (𝐾 ↑_m 𝐼)) → 𝑆 ∈ CRing)
154		simpr 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑚 ∈ (𝐾 ↑_m 𝐼)) → 𝑚 ∈ (𝐾 ↑_m 𝐼))
155		simplr 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑚 ∈ (𝐾 ↑_m 𝐼)) → 𝑏 ∈ 𝐷)
156	11, 12, 84, 85, 152, 153, 154, 155	evlsvvvallem 41859	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑚 ∈ (𝐾 ↑_m 𝐼)) → (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖)))) ∈ 𝐾)
157	156	fmpttd 7120	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖))))):(𝐾 ↑_m 𝐼)⟶𝐾)
158	14, 12, 51, 102, 101, 157	pwselbasr 41829	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖))))) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
159		evlsvvval.x	. . . . . . 7 ⊢ · = (._r‘𝑆)
160	14, 51, 102, 101, 151, 158, 159, 17	pwsmulrval 17472	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (((𝐾 ↑_m 𝐼) × {(𝐹‘𝑏)})(._r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))(𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖)))))) = (((𝐾 ↑_m 𝐼) × {(𝐹‘𝑏)}) ∘_f · (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖)))))))
161	150	ffnd 6718	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((𝐾 ↑_m 𝐼) × {(𝐹‘𝑏)}) Fn (𝐾 ↑_m 𝐼))
162		ovex 7449	. . . . . . . . 9 ⊢ (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖)))) ∈ V
163		eqid 2725	. . . . . . . . 9 ⊢ (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖))))) = (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖)))))
164	162, 163	fnmpti 6693	. . . . . . . 8 ⊢ (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖))))) Fn (𝐾 ↑_m 𝐼)
165	164	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖))))) Fn (𝐾 ↑_m 𝐼))
166		inidm 4213	. . . . . . 7 ⊢ ((𝐾 ↑_m 𝐼) ∩ (𝐾 ↑_m 𝐼)) = (𝐾 ↑_m 𝐼)
167		fvex 6905	. . . . . . . . 9 ⊢ (𝐹‘𝑏) ∈ V
168	167	fvconst2 7212	. . . . . . . 8 ⊢ (𝑙 ∈ (𝐾 ↑_m 𝐼) → (((𝐾 ↑_m 𝐼) × {(𝐹‘𝑏)})‘𝑙) = (𝐹‘𝑏))
169	168	adantl 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → (((𝐾 ↑_m 𝐼) × {(𝐹‘𝑏)})‘𝑙) = (𝐹‘𝑏))
170		fveq1 6891	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑙 → (𝑚‘𝑖) = (𝑙‘𝑖))
171	170	oveq2d 7432	. . . . . . . . . 10 ⊢ (𝑚 = 𝑙 → ((𝑏‘𝑖) ↑ (𝑚‘𝑖)) = ((𝑏‘𝑖) ↑ (𝑙‘𝑖)))
172	171	mpteq2dv 5245	. . . . . . . . 9 ⊢ (𝑚 = 𝑙 → (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖))) = (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖))))
173	172	oveq2d 7432	. . . . . . . 8 ⊢ (𝑚 = 𝑙 → (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖)))) = (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖)))))
174		simpr 483	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → 𝑙 ∈ (𝐾 ↑_m 𝐼))
175	20	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → 𝐼 ∈ 𝑉)
176	21	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → 𝑆 ∈ CRing)
177		simplr 767	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → 𝑏 ∈ 𝐷)
178	11, 12, 84, 85, 175, 176, 174, 177	evlsvvvallem 41859	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖)))) ∈ 𝐾)
179	163, 173, 174, 178	fvmptd3 7023	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → ((𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖)))))‘𝑙) = (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖)))))
180	161, 165, 101, 101, 166, 169, 179	offval 7691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (((𝐾 ↑_m 𝐼) × {(𝐹‘𝑏)}) ∘_f · (𝑚 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑚‘𝑖)))))) = (𝑙 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖)))))))
181	143, 160, 180	3eqtrd 2769	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝑏))(._r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))((mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) Σ_g (𝑏 ∘_f (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)))))) = (𝑙 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖)))))))
182	181	mpteq2dva 5243	. . . 4 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ (((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝑏))(._r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))((mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) Σ_g (𝑏 ∘_f (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥))))))) = (𝑏 ∈ 𝐷 ↦ (𝑙 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖))))))))
183	182	oveq2d 7432	. . 3 ⊢ (𝜑 → ((𝑆 ↑_s (𝐾 ↑_m 𝐼)) Σ_g (𝑏 ∈ 𝐷 ↦ (((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝑏))(._r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))((mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) Σ_g (𝑏 ∘_f (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)))))))) = ((𝑆 ↑_s (𝐾 ↑_m 𝐼)) Σ_g (𝑏 ∈ 𝐷 ↦ (𝑙 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖)))))))))
184		eqid 2725	. . . 4 ⊢ (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
185		ovexd 7451	. . . . 5 ⊢ (𝜑 → (ℕ₀ ↑_m 𝐼) ∈ V)
186	11, 185	rabexd 5330	. . . 4 ⊢ (𝜑 → 𝐷 ∈ V)
187	63	ringcmnd 20224	. . . 4 ⊢ (𝜑 → 𝑆 ∈ CMnd)
188	63	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑙 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑏 ∈ 𝐷)) → 𝑆 ∈ Ring)
189	148	adantrl 714	. . . . 5 ⊢ ((𝜑 ∧ (𝑙 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑏 ∈ 𝐷)) → (𝐹‘𝑏) ∈ 𝐾)
190		simpl 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑙 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑏 ∈ 𝐷)) → 𝜑)
191		simprr 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑙 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑏 ∈ 𝐷)) → 𝑏 ∈ 𝐷)
192		simprl 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑙 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑏 ∈ 𝐷)) → 𝑙 ∈ (𝐾 ↑_m 𝐼))
193	190, 191, 192, 178	syl21anc 836	. . . . 5 ⊢ ((𝜑 ∧ (𝑙 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑏 ∈ 𝐷)) → (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖)))) ∈ 𝐾)
194	12, 159, 188, 189, 193	ringcld 20203	. . . 4 ⊢ ((𝜑 ∧ (𝑙 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑏 ∈ 𝐷)) → ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖))))) ∈ 𝐾)
195	186	mptexd 7232	. . . . 5 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ (𝑙 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖))))))) ∈ V)
196		fvexd 6907	. . . . 5 ⊢ (𝜑 → (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∈ V)
197		funmpt 6586	. . . . . 6 ⊢ Fun (𝑏 ∈ 𝐷 ↦ (𝑙 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖)))))))
198	197	a1i 11	. . . . 5 ⊢ (𝜑 → Fun (𝑏 ∈ 𝐷 ↦ (𝑙 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖))))))))
199		eqid 2725	. . . . . 6 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
200	13	ovexi 7450	. . . . . . 7 ⊢ 𝑈 ∈ V
201	200	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ V)
202	9, 10, 199, 23, 201	mplelsfi 21944	. . . . 5 ⊢ (𝜑 → 𝐹 finSupp (0_g‘𝑈))
203		ssidd 3996	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 supp (0_g‘𝑈)) ⊆ (𝐹 supp (0_g‘𝑈)))
204		fvexd 6907	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0_g‘𝑈) ∈ V)
205	147, 203, 186, 204	suppssr 8199	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) → (𝐹‘𝑏) = (0_g‘𝑈))
206		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
207	13, 206	subrg0 20522	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (0_g‘𝑆) = (0_g‘𝑈))
208	22, 207	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0_g‘𝑆) = (0_g‘𝑈))
209	208	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) → (0_g‘𝑆) = (0_g‘𝑈))
210	205, 209	eqtr4d 2768	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) → (𝐹‘𝑏) = (0_g‘𝑆))
211	210	adantr 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → (𝐹‘𝑏) = (0_g‘𝑆))
212	211	oveq1d 7431	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖))))) = ((0_g‘𝑆) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖))))))
213	63	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → 𝑆 ∈ Ring)
214		eldifi 4119	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈))) → 𝑏 ∈ 𝐷)
215	214, 178	sylanl2 679	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖)))) ∈ 𝐾)
216	12, 159, 206, 213, 215	ringlzd 20235	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → ((0_g‘𝑆) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖))))) = (0_g‘𝑆))
217	212, 216	eqtrd 2765	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖))))) = (0_g‘𝑆))
218	217	mpteq2dva 5243	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) → (𝑙 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖)))))) = (𝑙 ∈ (𝐾 ↑_m 𝐼) ↦ (0_g‘𝑆)))
219		fconstmpt 5734	. . . . . . . . 9 ⊢ ((𝐾 ↑_m 𝐼) × {(0_g‘𝑆)}) = (𝑙 ∈ (𝐾 ↑_m 𝐼) ↦ (0_g‘𝑆))
220	187	cmnmndd 19763	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ Mnd)
221	14, 206	pws0g 18729	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Mnd ∧ (𝐾 ↑_m 𝐼) ∈ V) → ((𝐾 ↑_m 𝐼) × {(0_g‘𝑆)}) = (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
222	220, 64, 221	syl2anc 582	. . . . . . . . 9 ⊢ (𝜑 → ((𝐾 ↑_m 𝐼) × {(0_g‘𝑆)}) = (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
223	219, 222	eqtr3id 2779	. . . . . . . 8 ⊢ (𝜑 → (𝑙 ∈ (𝐾 ↑_m 𝐼) ↦ (0_g‘𝑆)) = (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
224	223	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) → (𝑙 ∈ (𝐾 ↑_m 𝐼) ↦ (0_g‘𝑆)) = (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
225	218, 224	eqtrd 2765	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0_g‘𝑈)))) → (𝑙 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖)))))) = (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
226	225, 186	suppss2 8204	. . . . 5 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ (𝑙 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖))))))) supp (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))) ⊆ (𝐹 supp (0_g‘𝑈)))
227	195, 196, 198, 202, 226	fsuppsssuppgd 9405	. . . 4 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ (𝑙 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖))))))) finSupp (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
228	14, 12, 184, 64, 186, 187, 194, 227	pwsgsum 19941	. . 3 ⊢ (𝜑 → ((𝑆 ↑_s (𝐾 ↑_m 𝐼)) Σ_g (𝑏 ∈ 𝐷 ↦ (𝑙 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖)))))))) = (𝑙 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑆 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖)))))))))
229	24, 183, 228	3eqtrd 2769	. 2 ⊢ (𝜑 → (𝑄‘𝐹) = (𝑙 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑆 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝑙‘𝑖)))))))))
230		evlsvvval.a	. 2 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))
231		ovexd 7451	. 2 ⊢ (𝜑 → (𝑆 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))) ∈ V)
232	7, 229, 230, 231	fvmptd4 7024	1 ⊢ (𝜑 → ((𝑄‘𝐹)‘𝐴) = (𝑆 Σ_g (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {crab 3419 Vcvv 3463 ∖ cdif 3936 ⊆ wss 3939 {csn 4624 class class class wbr 5143 ↦ cmpt 5226 × cxp 5670 ^◡ccnv 5671 “ cima 5675 Fun wfun 6537 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7416 ∘_f cof 7680 supp csupp 8163 ↑_m cmap 8843 Fincfn 8962 finSupp cfsupp 9385 ℂcc 11136 0cc0 11138 ℕcn 12242 ℕ₀cn0 12502 Basecbs 17179 ↾_s cress 17208 ._rcmulr 17233 0_gc0g 17420 Σ_g cgsu 17421 ↑_s cpws 17427 Mndcmnd 18693 ._gcmg 19027 mulGrpcmgp 20078 1_rcur 20125 Ringcrg 20177 CRingccrg 20178 SubRingcsubrg 20510 mPoly cmpl 21843 evalSub ces 22023

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-ofr 7683 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-fzo 13660 df-seq 13999 df-hash 14322 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-hom 17256 df-cco 17257 df-0g 17422 df-gsum 17423 df-prds 17428 df-pws 17430 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-mulg 19028 df-subg 19082 df-ghm 19172 df-cntz 19272 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-srg 20131 df-ring 20179 df-cring 20180 df-rhm 20415 df-subrng 20487 df-subrg 20512 df-lmod 20749 df-lss 20820 df-lsp 20860 df-assa 21791 df-asp 21792 df-ascl 21793 df-psr 21846 df-mvr 21847 df-mpl 21848 df-evls 22025

This theorem is referenced by: evlsbagval 41864 evlvvval 41871 evlsmhpvvval 41893

W3C validator