evlsbagval - Mathbox for Steven Nguyen

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Theorem evlsbagval 40198

Description: Polynomial evaluation builder for a bag of variables. EDITORIAL: This theorem should stay in my mathbox until there's another use, since 0 and 1 using 𝑈 instead of 𝑆 is convenient for its sole use case mhphf 40208, but may not be convenient for other uses. (Contributed by SN, 29-Jul-2024.)

**Hypotheses**
Ref	Expression
evlsbagval.q	⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
evlsbagval.p	⊢ 𝑃 = (𝐼 mPoly 𝑈)
evlsbagval.u	⊢ 𝑈 = (𝑆 ↾_s 𝑅)
evlsbagval.w	⊢ 𝑊 = (Base‘𝑃)
evlsbagval.k	⊢ 𝐾 = (Base‘𝑆)
evlsbagval.m	⊢ 𝑀 = (mulGrp‘𝑆)
evlsbagval.e	⊢ ↑ = (._g‘𝑀)
evlsbagval.z	⊢ 0 = (0_g‘𝑈)
evlsbagval.o	⊢ 1 = (1_r‘𝑈)
evlsbagval.d	⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
evlsbagval.f	⊢ 𝐹 = (𝑠 ∈ 𝐷 ↦ if(𝑠 = 𝐵, 1 , 0 ))
evlsbagval.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
evlsbagval.s	⊢ (𝜑 → 𝑆 ∈ CRing)
evlsbagval.r	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
evlsbagval.a	⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))
evlsbagval.b	⊢ (𝜑 → 𝐵 ∈ 𝐷)

**Assertion**
Ref	Expression
evlsbagval	⊢ (𝜑 → (𝐹 ∈ 𝑊 ∧ ((𝑄‘𝐹)‘𝐴) = (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣))))))

Distinct variable groups: 0 ,𝑠 1 ,𝑠 𝑣,𝐴 𝑣,𝐵 𝐵,ℎ 𝐵,𝑠 𝑣,𝐷 𝐷,𝑠 𝑣,𝐼 ℎ,𝐼 𝑣,𝐾 𝑅,𝑠 𝑣,𝑆 𝑈,ℎ 𝑈,𝑠 𝜑,𝑣 𝜑,𝑠 Allowed substitution hints: 𝜑(ℎ) 𝐴(ℎ,𝑠) 𝐷(ℎ) 𝑃(𝑣,ℎ,𝑠) 𝑄(𝑣,ℎ,𝑠) 𝑅(𝑣,ℎ) 𝑆(ℎ,𝑠) 𝑈(𝑣) 1 (𝑣,ℎ) ↑ (𝑣,ℎ,𝑠) 𝐹(𝑣,ℎ,𝑠) 𝐼(𝑠) 𝐾(ℎ,𝑠) 𝑀(𝑣,ℎ,𝑠) 𝑉(𝑣,ℎ,𝑠) 𝑊(𝑣,ℎ,𝑠) 0 (𝑣,ℎ)

Proof of Theorem evlsbagval Dummy variables 𝑥 𝑔 𝑙 𝑎 𝑏 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6771	. . . . 5 ⊢ (𝜑 → (Base‘𝑈) ∈ V)
2		evlsbagval.d	. . . . . 6 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
3		ovexd 7290	. . . . . 6 ⊢ (𝜑 → (ℕ₀ ↑_m 𝐼) ∈ V)
4	2, 3	rabexd 5252	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ V)
5		evlsbagval.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
6		evlsbagval.u	. . . . . . . . . . 11 ⊢ 𝑈 = (𝑆 ↾_s 𝑅)
7	6	subrgring 19942	. . . . . . . . . 10 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
8	5, 7	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ Ring)
9		eqid 2738	. . . . . . . . . 10 ⊢ (Base‘𝑈) = (Base‘𝑈)
10		evlsbagval.o	. . . . . . . . . 10 ⊢ 1 = (1_r‘𝑈)
11	9, 10	ringidcl 19722	. . . . . . . . 9 ⊢ (𝑈 ∈ Ring → 1 ∈ (Base‘𝑈))
12	8, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ (Base‘𝑈))
13		evlsbagval.z	. . . . . . . . . 10 ⊢ 0 = (0_g‘𝑈)
14	9, 13	ring0cl 19723	. . . . . . . . 9 ⊢ (𝑈 ∈ Ring → 0 ∈ (Base‘𝑈))
15	8, 14	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (Base‘𝑈))
16	12, 15	ifcld 4502	. . . . . . 7 ⊢ (𝜑 → if(𝑠 = 𝐵, 1 , 0 ) ∈ (Base‘𝑈))
17	16	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → if(𝑠 = 𝐵, 1 , 0 ) ∈ (Base‘𝑈))
18		evlsbagval.f	. . . . . 6 ⊢ 𝐹 = (𝑠 ∈ 𝐷 ↦ if(𝑠 = 𝐵, 1 , 0 ))
19	17, 18	fmptd 6970	. . . . 5 ⊢ (𝜑 → 𝐹:𝐷⟶(Base‘𝑈))
20	1, 4, 19	elmapdd 40142	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((Base‘𝑈) ↑_m 𝐷))
21		eqid 2738	. . . . 5 ⊢ (𝐼 mPwSer 𝑈) = (𝐼 mPwSer 𝑈)
22		eqid 2738	. . . . 5 ⊢ (Base‘(𝐼 mPwSer 𝑈)) = (Base‘(𝐼 mPwSer 𝑈))
23		evlsbagval.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
24	21, 9, 2, 22, 23	psrbas 21057	. . . 4 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑈)) = ((Base‘𝑈) ↑_m 𝐷))
25	20, 24	eleqtrrd 2842	. . 3 ⊢ (𝜑 → 𝐹 ∈ (Base‘(𝐼 mPwSer 𝑈)))
26	4, 15, 18	sniffsupp 9089	. . 3 ⊢ (𝜑 → 𝐹 finSupp 0 )
27		evlsbagval.p	. . . 4 ⊢ 𝑃 = (𝐼 mPoly 𝑈)
28		evlsbagval.w	. . . 4 ⊢ 𝑊 = (Base‘𝑃)
29	27, 21, 22, 13, 28	mplelbas 21109	. . 3 ⊢ (𝐹 ∈ 𝑊 ↔ (𝐹 ∈ (Base‘(𝐼 mPwSer 𝑈)) ∧ 𝐹 finSupp 0 ))
30	25, 26, 29	sylanbrc 582	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝑊)
31		fveq1 6755	. . . . . 6 ⊢ (𝑔 = 𝐴 → (𝑔‘𝑣) = (𝐴‘𝑣))
32	31	oveq2d 7271	. . . . 5 ⊢ (𝑔 = 𝐴 → ((𝐵‘𝑣) ↑ (𝑔‘𝑣)) = ((𝐵‘𝑣) ↑ (𝐴‘𝑣)))
33	32	mpteq2dv 5172	. . . 4 ⊢ (𝑔 = 𝐴 → (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))) = (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣))))
34	33	oveq2d 7271	. . 3 ⊢ (𝑔 = 𝐴 → (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)))) = (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣)))))
35		fveq1 6755	. . . . . . . . 9 ⊢ (𝑝 = 𝐹 → (𝑝‘𝑏) = (𝐹‘𝑏))
36	35	fveq2d 6760	. . . . . . . 8 ⊢ (𝑝 = 𝐹 → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝑝‘𝑏)) = ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝑏)))
37	36	oveq1d 7270	. . . . . . 7 ⊢ (𝑝 = 𝐹 → (((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝑝‘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)))))) = (((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)))))))
38	37	mpteq2dv 5172	. . . . . 6 ⊢ (𝑝 = 𝐹 → (𝑏 ∈ 𝐷 ↦ (((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝑝‘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 ↦ (((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))))))))
39	38	oveq2d 7271	. . . . 5 ⊢ (𝑝 = 𝐹 → ((𝑆 ↑_s (𝐾 ↑_m 𝐼)) Σ_g (𝑏 ∈ 𝐷 ↦ (((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝑝‘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 ↦ (((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)))))))))
40		evlsbagval.q	. . . . . 6 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
41		evlsbagval.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝑆)
42		eqid 2738	. . . . . 6 ⊢ (𝑆 ↑_s (𝐾 ↑_m 𝐼)) = (𝑆 ↑_s (𝐾 ↑_m 𝐼))
43		eqid 2738	. . . . . 6 ⊢ (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
44		eqid 2738	. . . . . 6 ⊢ (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))) = (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
45		eqid 2738	. . . . . 6 ⊢ (._r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (._r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
46		eqid 2738	. . . . . 6 ⊢ (𝑝 ∈ 𝑊 ↦ ((𝑆 ↑_s (𝐾 ↑_m 𝐼)) Σ_g (𝑏 ∈ 𝐷 ↦ (((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝑝‘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))))))))) = (𝑝 ∈ 𝑊 ↦ ((𝑆 ↑_s (𝐾 ↑_m 𝐼)) Σ_g (𝑏 ∈ 𝐷 ↦ (((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝑝‘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)))))))))
47		eqid 2738	. . . . . 6 ⊢ (𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥})) = (𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))
48		eqid 2738	. . . . . 6 ⊢ (𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)))
49		evlsbagval.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
50	40, 27, 28, 2, 41, 6, 42, 43, 44, 45, 46, 47, 48, 23, 49, 5	evlsval3 40195	. . . . 5 ⊢ (𝜑 → 𝑄 = (𝑝 ∈ 𝑊 ↦ ((𝑆 ↑_s (𝐾 ↑_m 𝐼)) Σ_g (𝑏 ∈ 𝐷 ↦ (((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝑝‘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))))))))))
51		ovexd 7290	. . . . 5 ⊢ (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)))))))) ∈ V)
52	39, 50, 30, 51	fvmptd4 40136	. . . 4 ⊢ (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)))))))))
53		eqid 2738	. . . . . 6 ⊢ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
54		eqid 2738	. . . . . 6 ⊢ (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
55	49	crngringd 19711	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Ring)
56		ovexd 7290	. . . . . . . 8 ⊢ (𝜑 → (𝐾 ↑_m 𝐼) ∈ V)
57	42	pwsring 19769	. . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ (𝐾 ↑_m 𝐼) ∈ V) → (𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ Ring)
58	55, 56, 57	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ Ring)
59	58	ringcmnd 40171	. . . . . 6 ⊢ (𝜑 → (𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ CMnd)
60	58	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ Ring)
61	49	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝑅) → 𝑆 ∈ CRing)
62		ovexd 7290	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝑅) → (𝐾 ↑_m 𝐼) ∈ V)
63	41	subrgss 19940	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐾)
64	5, 63	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ⊆ 𝐾)
65	64	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑅 ⊆ 𝐾)
66	65	sselda 3917	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝑅) → 𝑥 ∈ 𝐾)
67		fconst6g 6647	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐾 → ((𝐾 ↑_m 𝐼) × {𝑥}):(𝐾 ↑_m 𝐼)⟶𝐾)
68	66, 67	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝑅) → ((𝐾 ↑_m 𝐼) × {𝑥}):(𝐾 ↑_m 𝐼)⟶𝐾)
69	42, 41, 53, 61, 62, 68	pwselbasr 40191	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝑅) → ((𝐾 ↑_m 𝐼) × {𝑥}) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
70	69	fmpttd 6971	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥})):𝑅⟶(Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
71		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (1_r‘𝑆) = (1_r‘𝑆)
72	6, 71	subrg1 19949	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (1_r‘𝑆) = (1_r‘𝑈))
73	5, 72	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1_r‘𝑆) = (1_r‘𝑈))
74	71	subrg1cl 19947	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (1_r‘𝑆) ∈ 𝑅)
75	5, 74	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1_r‘𝑆) ∈ 𝑅)
76	73, 75	eqeltrrd 2840	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1_r‘𝑈) ∈ 𝑅)
77	10, 76	eqeltrid 2843	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ 𝑅)
78	6	subrgbas 19948	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 = (Base‘𝑈))
79	5, 78	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 = (Base‘𝑈))
80	15, 79	eleqtrrd 2842	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ 𝑅)
81	77, 80	ifcld 4502	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑠 = 𝐵, 1 , 0 ) ∈ 𝑅)
82	81	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → if(𝑠 = 𝐵, 1 , 0 ) ∈ 𝑅)
83	82, 18	fmptd 6970	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐷⟶𝑅)
84	83	ffvelrnda 6943	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘𝑏) ∈ 𝑅)
85	70, 84	ffvelrnd 6944	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝑏)) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
86	43, 53	mgpbas 19641	. . . . . . . . 9 ⊢ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (Base‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
87	42	pwscrng 19771	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ CRing ∧ (𝐾 ↑_m 𝐼) ∈ V) → (𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ CRing)
88	49, 56, 87	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ CRing)
89	43	crngmgp 19706	. . . . . . . . . . 11 ⊢ ((𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ CRing → (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∈ CMnd)
90	88, 89	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∈ CMnd)
91	90	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∈ CMnd)
92		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ 𝐷)
93	49	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ CRing)
94		ovexd 7290	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝐼) → (𝐾 ↑_m 𝐼) ∈ V)
95		elmapi 8595	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (𝐾 ↑_m 𝐼) → 𝑎:𝐼⟶𝐾)
96	95	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → 𝑎:𝐼⟶𝐾)
97		simplr 765	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → 𝑥 ∈ 𝐼)
98	96, 97	ffvelrnd 6944	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → (𝑎‘𝑥) ∈ 𝐾)
99	98	fmpttd 6971	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝐼) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)):(𝐾 ↑_m 𝐼)⟶𝐾)
100	42, 41, 53, 93, 94, 99	pwselbasr 40191	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝐼) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
101	100	fmpttd 6971	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥))):𝐼⟶(Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
102	2, 86, 44, 91, 92, 101	psrbagev2 21197	. . . . . . . 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))))) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
103	53, 45, 60, 85, 102	ringcld 40166	. . . . . . 7 ⊢ ((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)))))) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
104	103	fmpttd 6971	. . . . . 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))))))):𝐷⟶(Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
105		eqeq1 2742	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑏 → (𝑠 = 𝐵 ↔ 𝑏 = 𝐵))
106	105	ifbid 4479	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑏 → if(𝑠 = 𝐵, 1 , 0 ) = if(𝑏 = 𝐵, 1 , 0 ))
107		eldifi 4057	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ (𝐷 ∖ {𝐵}) → 𝑏 ∈ 𝐷)
108	107	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → 𝑏 ∈ 𝐷)
109	10	fvexi 6770	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
110	13	fvexi 6770	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
111	109, 110	ifex 4506	. . . . . . . . . . . . . 14 ⊢ if(𝑏 = 𝐵, 1 , 0 ) ∈ V
112	111	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → if(𝑏 = 𝐵, 1 , 0 ) ∈ V)
113	18, 106, 108, 112	fvmptd3 6880	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → (𝐹‘𝑏) = if(𝑏 = 𝐵, 1 , 0 ))
114		eldifsnneq 4721	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ (𝐷 ∖ {𝐵}) → ¬ 𝑏 = 𝐵)
115	114	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → ¬ 𝑏 = 𝐵)
116	115	iffalsed 4467	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → if(𝑏 = 𝐵, 1 , 0 ) = 0 )
117	113, 116	eqtrd 2778	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → (𝐹‘𝑏) = 0 )
118	117	fveq2d 6760	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝑏)) = ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘ 0 ))
119		sneq 4568	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → {𝑥} = { 0 })
120	119	xpeq2d 5610	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → ((𝐾 ↑_m 𝐼) × {𝑥}) = ((𝐾 ↑_m 𝐼) × { 0 }))
121	80	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → 0 ∈ 𝑅)
122		ovex 7288	. . . . . . . . . . . . 13 ⊢ (𝐾 ↑_m 𝐼) ∈ V
123		snex 5349	. . . . . . . . . . . . 13 ⊢ { 0 } ∈ V
124	122, 123	xpex 7581	. . . . . . . . . . . 12 ⊢ ((𝐾 ↑_m 𝐼) × { 0 }) ∈ V
125	124	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → ((𝐾 ↑_m 𝐼) × { 0 }) ∈ V)
126	47, 120, 121, 125	fvmptd3 6880	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘ 0 ) = ((𝐾 ↑_m 𝐼) × { 0 }))
127		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
128	6, 127	subrg0 19946	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (0_g‘𝑆) = (0_g‘𝑈))
129	5, 128	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0_g‘𝑆) = (0_g‘𝑈))
130	129, 13	eqtr4di 2797	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘𝑆) = 0 )
131	130	sneqd 4570	. . . . . . . . . . . . 13 ⊢ (𝜑 → {(0_g‘𝑆)} = { 0 })
132	131	xpeq2d 5610	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾 ↑_m 𝐼) × {(0_g‘𝑆)}) = ((𝐾 ↑_m 𝐼) × { 0 }))
133	55	ringgrpd 19707	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ Grp)
134	133	grpmndd 18504	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ Mnd)
135	42, 127	pws0g 18336	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ Mnd ∧ (𝐾 ↑_m 𝐼) ∈ V) → ((𝐾 ↑_m 𝐼) × {(0_g‘𝑆)}) = (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
136	134, 56, 135	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾 ↑_m 𝐼) × {(0_g‘𝑆)}) = (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
137	132, 136	eqtr3d 2780	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐾 ↑_m 𝐼) × { 0 }) = (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
138	137	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → ((𝐾 ↑_m 𝐼) × { 0 }) = (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
139	118, 126, 138	3eqtrd 2782	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝑏)) = (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
140	139	oveq1d 7270	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → (((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)))))) = ((0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))(._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)))))))
141	90	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∈ CMnd)
142	49	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ CRing)
143		ovexd 7290	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) ∧ 𝑥 ∈ 𝐼) → (𝐾 ↑_m 𝐼) ∈ V)
144	95	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) ∧ 𝑥 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → 𝑎:𝐼⟶𝐾)
145		simplr 765	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) ∧ 𝑥 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → 𝑥 ∈ 𝐼)
146	144, 145	ffvelrnd 6944	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) ∧ 𝑥 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → (𝑎‘𝑥) ∈ 𝐾)
147	146	fmpttd 6971	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) ∧ 𝑥 ∈ 𝐼) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)):(𝐾 ↑_m 𝐼)⟶𝐾)
148	42, 41, 53, 142, 143, 147	pwselbasr 40191	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) ∧ 𝑥 ∈ 𝐼) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
149	148	fmpttd 6971	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → (𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥))):𝐼⟶(Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
150	2, 86, 44, 141, 108, 149	psrbagev2 21197	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → ((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))))) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
151	53, 45, 54	ringlz 19741	. . . . . . . . 9 ⊢ (((𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ Ring ∧ ((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))))) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))) → ((0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))(._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)))))) = (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
152	58, 150, 151	syl2an2r 681	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → ((0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))(._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)))))) = (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
153	140, 152	eqtrd 2778	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → (((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)))))) = (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
154	153, 4	suppss2 7987	. . . . . 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))))))) supp (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))) ⊆ {𝐵})
155	4	mptexd 7082	. . . . . . 7 ⊢ (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))))))) ∈ V)
156		fvexd 6771	. . . . . . 7 ⊢ (𝜑 → (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∈ V)
157		funmpt 6456	. . . . . . . 8 ⊢ Fun (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)))))))
158	157	a1i 11	. . . . . . 7 ⊢ (𝜑 → Fun (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))))))))
159		snfi 8788	. . . . . . . . 9 ⊢ {𝐵} ∈ Fin
160	159	a1i 11	. . . . . . . 8 ⊢ (𝜑 → {𝐵} ∈ Fin)
161	160, 154	ssfid 8971	. . . . . . 7 ⊢ (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))))))) supp (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))) ∈ Fin)
162	155, 156, 158, 161	isfsuppd 40143	. . . . . 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))))))) finSupp (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
163	53, 54, 59, 4, 104, 154, 162	gsumres 19429	. . . . 5 ⊢ (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))))))) ↾ {𝐵})) = ((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)))))))))
164		evlsbagval.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
165	164	snssd 4739	. . . . . . 7 ⊢ (𝜑 → {𝐵} ⊆ 𝐷)
166	165	resmptd 5937	. . . . . 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))))))) ↾ {𝐵}) = (𝑏 ∈ {𝐵} ↦ (((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))))))))
167	166	oveq2d 7271	. . . . 5 ⊢ (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))))))) ↾ {𝐵})) = ((𝑆 ↑_s (𝐾 ↑_m 𝐼)) Σ_g (𝑏 ∈ {𝐵} ↦ (((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)))))))))
168	163, 167	eqtr3d 2780	. . . 4 ⊢ (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 ∈ {𝐵} ↦ (((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)))))))))
169	58	ringgrpd 19707	. . . . . . 7 ⊢ (𝜑 → (𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ Grp)
170	169	grpmndd 18504	. . . . . 6 ⊢ (𝜑 → (𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ Mnd)
171		iftrue 4462	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝐵 → if(𝑠 = 𝐵, 1 , 0 ) = 1 )
172	109	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ V)
173	18, 171, 164, 172	fvmptd3 6880	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘𝐵) = 1 )
174	173	fveq2d 6760	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝐵)) = ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘ 1 ))
175	10, 73	eqtr4id 2798	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 = (1_r‘𝑆))
176	175	fveq2d 6760	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘ 1 ) = ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(1_r‘𝑆)))
177		sneq 4568	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1_r‘𝑆) → {𝑥} = {(1_r‘𝑆)})
178	177	xpeq2d 5610	. . . . . . . . . . . 12 ⊢ (𝑥 = (1_r‘𝑆) → ((𝐾 ↑_m 𝐼) × {𝑥}) = ((𝐾 ↑_m 𝐼) × {(1_r‘𝑆)}))
179		snex 5349	. . . . . . . . . . . . . 14 ⊢ {(1_r‘𝑆)} ∈ V
180	179	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → {(1_r‘𝑆)} ∈ V)
181	56, 180	xpexd 7579	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾 ↑_m 𝐼) × {(1_r‘𝑆)}) ∈ V)
182	47, 178, 75, 181	fvmptd3 6880	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(1_r‘𝑆)) = ((𝐾 ↑_m 𝐼) × {(1_r‘𝑆)}))
183	176, 182	eqtrd 2778	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘ 1 ) = ((𝐾 ↑_m 𝐼) × {(1_r‘𝑆)}))
184	42, 71	pws1 19770	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Ring ∧ (𝐾 ↑_m 𝐼) ∈ V) → ((𝐾 ↑_m 𝐼) × {(1_r‘𝑆)}) = (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
185	55, 56, 184	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾 ↑_m 𝐼) × {(1_r‘𝑆)}) = (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
186	174, 183, 185	3eqtrd 2782	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝐵)) = (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
187	186	oveq1d 7270	. . . . . . . 8 ⊢ (𝜑 → (((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)))))) = ((1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))(._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)))))))
188		eqid 2738	. . . . . . . . 9 ⊢ (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
189	49	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ CRing)
190		ovexd 7290	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐾 ↑_m 𝐼) ∈ V)
191	95	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → 𝑎:𝐼⟶𝐾)
192		simplr 765	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → 𝑥 ∈ 𝐼)
193	191, 192	ffvelrnd 6944	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → (𝑎‘𝑥) ∈ 𝐾)
194	193	fmpttd 6971	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)):(𝐾 ↑_m 𝐼)⟶𝐾)
195	42, 41, 53, 189, 190, 194	pwselbasr 40191	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
196	195	fmpttd 6971	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥))):𝐼⟶(Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
197	2, 86, 44, 90, 164, 196	psrbagev2 21197	. . . . . . . . 9 ⊢ (𝜑 → ((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))))) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
198	53, 45, 188, 58, 197	ringlidmd 40168	. . . . . . . 8 ⊢ (𝜑 → ((1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))(._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)))))) = ((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))))))
199	187, 198	eqtrd 2778	. . . . . . 7 ⊢ (𝜑 → (((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)))))) = ((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))))))
200	199, 197	eqeltrd 2839	. . . . . 6 ⊢ (𝜑 → (((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)))))) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
201		2fveq3 6761	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝑏)) = ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝐵)))
202		oveq1 7262	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑏 ∘_f (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)))) = (𝐵 ∘_f (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)))))
203	202	oveq2d 7271	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → ((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 (𝐵 ∘_f (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥))))))
204	201, 203	oveq12d 7273	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (((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)))))) = (((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)))))))
205	53, 204	gsumsn 19470	. . . . . 6 ⊢ (((𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ Mnd ∧ 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)))))) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))) → ((𝑆 ↑_s (𝐾 ↑_m 𝐼)) Σ_g (𝑏 ∈ {𝐵} ↦ (((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)))))))) = (((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)))))))
206	170, 164, 200, 205	syl3anc 1369	. . . . 5 ⊢ (𝜑 → ((𝑆 ↑_s (𝐾 ↑_m 𝐼)) Σ_g (𝑏 ∈ {𝐵} ↦ (((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)))))))) = (((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)))))))
207	2	psrbagf 21031	. . . . . . . . . . 11 ⊢ (𝐵 ∈ 𝐷 → 𝐵:𝐼⟶ℕ₀)
208	164, 207	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵:𝐼⟶ℕ₀)
209	208	ffnd 6585	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 Fn 𝐼)
210	122	mptex 7081	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)) ∈ V
211	210, 48	fnmpti 6560	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥))) Fn 𝐼
212	211	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥))) Fn 𝐼)
213		inidm 4149	. . . . . . . . 9 ⊢ (𝐼 ∩ 𝐼) = 𝐼
214		eqidd 2739	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → (𝐵‘𝑣) = (𝐵‘𝑣))
215		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑣 → (𝑎‘𝑥) = (𝑎‘𝑣))
216	215	mpteq2dv 5172	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)) = (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣)))
217		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → 𝑣 ∈ 𝐼)
218	122	mptex 7081	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣)) ∈ V
219	218	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣)) ∈ V)
220	48, 216, 217, 219	fvmptd3 6880	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → ((𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)))‘𝑣) = (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣)))
221	209, 212, 23, 23, 213, 214, 220	offval 7520	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ∘_f (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)))) = (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣)(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣)))))
222	49	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → 𝑆 ∈ CRing)
223		ovexd 7290	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → (𝐾 ↑_m 𝐼) ∈ V)
224	43	ringmgp 19704	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ Ring → (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∈ Mnd)
225	58, 224	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∈ Mnd)
226	225	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∈ Mnd)
227	208	ffvelrnda 6943	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → (𝐵‘𝑣) ∈ ℕ₀)
228	95	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → 𝑎:𝐼⟶𝐾)
229		simplr 765	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → 𝑣 ∈ 𝐼)
230	228, 229	ffvelrnd 6944	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → (𝑎‘𝑣) ∈ 𝐾)
231	230	fmpttd 6971	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣)):(𝐾 ↑_m 𝐼)⟶𝐾)
232	42, 41, 53, 222, 223, 231	pwselbasr 40191	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣)) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
233	86, 44	mulgnn0cl 18635	. . . . . . . . . . . . 13 ⊢ (((mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∈ Mnd ∧ (𝐵‘𝑣) ∈ ℕ₀ ∧ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣)) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))) → ((𝐵‘𝑣)(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣))) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
234	226, 227, 232, 233	syl3anc 1369	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → ((𝐵‘𝑣)(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣))) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
235	42, 41, 53, 222, 223, 234	pwselbas 17117	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → ((𝐵‘𝑣)(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣))):(𝐾 ↑_m 𝐼)⟶𝐾)
236	235	ffnd 6585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → ((𝐵‘𝑣)(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣))) Fn (𝐾 ↑_m 𝐼))
237		ovex 7288	. . . . . . . . . . . 12 ⊢ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)) ∈ V
238		eqid 2738	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))) = (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)))
239	237, 238	fnmpti 6560	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))) Fn (𝐾 ↑_m 𝐼)
240	239	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))) Fn (𝐾 ↑_m 𝐼))
241		eqid 2738	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣)) = (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣))
242		fveq1 6755	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑙 → (𝑎‘𝑣) = (𝑙‘𝑣))
243		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → 𝑙 ∈ (𝐾 ↑_m 𝐼))
244		fvexd 6771	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → (𝑙‘𝑣) ∈ V)
245	241, 242, 243, 244	fvmptd3 6880	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → ((𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣))‘𝑙) = (𝑙‘𝑣))
246	245	oveq2d 7271	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → ((𝐵‘𝑣) ↑ ((𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣))‘𝑙)) = ((𝐵‘𝑣) ↑ (𝑙‘𝑣)))
247		evlsbagval.m	. . . . . . . . . . . 12 ⊢ 𝑀 = (mulGrp‘𝑆)
248		evlsbagval.e	. . . . . . . . . . . 12 ⊢ ↑ = (._g‘𝑀)
249	55	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → 𝑆 ∈ Ring)
250		ovexd 7290	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → (𝐾 ↑_m 𝐼) ∈ V)
251	227	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → (𝐵‘𝑣) ∈ ℕ₀)
252	232	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣)) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
253	42, 53, 43, 247, 44, 248, 249, 250, 251, 252, 243	pwsexpg 40193	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → (((𝐵‘𝑣)(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣)))‘𝑙) = ((𝐵‘𝑣) ↑ ((𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣))‘𝑙)))
254		fveq1 6755	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑙 → (𝑔‘𝑣) = (𝑙‘𝑣))
255	254	oveq2d 7271	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑙 → ((𝐵‘𝑣) ↑ (𝑔‘𝑣)) = ((𝐵‘𝑣) ↑ (𝑙‘𝑣)))
256		ovexd 7290	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → ((𝐵‘𝑣) ↑ (𝑙‘𝑣)) ∈ V)
257	238, 255, 243, 256	fvmptd3 6880	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → ((𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)))‘𝑙) = ((𝐵‘𝑣) ↑ (𝑙‘𝑣)))
258	246, 253, 257	3eqtr4d 2788	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → (((𝐵‘𝑣)(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣)))‘𝑙) = ((𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)))‘𝑙))
259	236, 240, 258	eqfnfvd 6894	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → ((𝐵‘𝑣)(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣))) = (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))))
260	259	mpteq2dva 5170	. . . . . . . 8 ⊢ (𝜑 → (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣)(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣)))) = (𝑣 ∈ 𝐼 ↦ (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)))))
261	221, 260	eqtrd 2778	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∘_f (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)))) = (𝑣 ∈ 𝐼 ↦ (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)))))
262	261	oveq2d 7271	. . . . . 6 ⊢ (𝜑 → ((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 (𝑣 ∈ 𝐼 ↦ (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))))))
263	247	crngmgp 19706	. . . . . . . . . . 11 ⊢ (𝑆 ∈ CRing → 𝑀 ∈ CMnd)
264	49, 263	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ CMnd)
265	264	cmnmndd 19324	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ Mnd)
266	265	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑣 ∈ 𝐼)) → 𝑀 ∈ Mnd)
267	227	adantrl 712	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑣 ∈ 𝐼)) → (𝐵‘𝑣) ∈ ℕ₀)
268		elmapi 8595	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝐾 ↑_m 𝐼) → 𝑔:𝐼⟶𝐾)
269	268	ad2antrl 724	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑣 ∈ 𝐼)) → 𝑔:𝐼⟶𝐾)
270		simprr 769	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑣 ∈ 𝐼)) → 𝑣 ∈ 𝐼)
271	269, 270	ffvelrnd 6944	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑣 ∈ 𝐼)) → (𝑔‘𝑣) ∈ 𝐾)
272	247, 41	mgpbas 19641	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝑀)
273	272, 248	mulgnn0cl 18635	. . . . . . . 8 ⊢ ((𝑀 ∈ Mnd ∧ (𝐵‘𝑣) ∈ ℕ₀ ∧ (𝑔‘𝑣) ∈ 𝐾) → ((𝐵‘𝑣) ↑ (𝑔‘𝑣)) ∈ 𝐾)
274	266, 267, 271, 273	syl3anc 1369	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑣 ∈ 𝐼)) → ((𝐵‘𝑣) ↑ (𝑔‘𝑣)) ∈ 𝐾)
275	23	mptexd 7082	. . . . . . . 8 ⊢ (𝜑 → (𝑣 ∈ 𝐼 ↦ (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)))) ∈ V)
276		fvexd 6771	. . . . . . . 8 ⊢ (𝜑 → (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∈ V)
277		funmpt 6456	. . . . . . . . 9 ⊢ Fun (𝑣 ∈ 𝐼 ↦ (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))))
278	277	a1i 11	. . . . . . . 8 ⊢ (𝜑 → Fun (𝑣 ∈ 𝐼 ↦ (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)))))
279	2	psrbagfsupp 21033	. . . . . . . . . . 11 ⊢ (𝐵 ∈ 𝐷 → 𝐵 finSupp 0)
280	164, 279	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 finSupp 0)
281	280	fsuppimpd 9065	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 supp 0) ∈ Fin)
282		ssidd 3940	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 supp 0) ⊆ (𝐵 supp 0))
283		c0ex 10900	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ V
284	283	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ V)
285	208, 282, 23, 284	suppssr 7983	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) → (𝐵‘𝑣) = 0)
286	285	oveq1d 7270	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) → ((𝐵‘𝑣) ↑ (𝑔‘𝑣)) = (0 ↑ (𝑔‘𝑣)))
287	286	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) ∧ 𝑔 ∈ (𝐾 ↑_m 𝐼)) → ((𝐵‘𝑣) ↑ (𝑔‘𝑣)) = (0 ↑ (𝑔‘𝑣)))
288	268	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) ∧ 𝑔 ∈ (𝐾 ↑_m 𝐼)) → 𝑔:𝐼⟶𝐾)
289		eldifi 4057	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ (𝐼 ∖ (𝐵 supp 0)) → 𝑣 ∈ 𝐼)
290	289	ad2antlr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) ∧ 𝑔 ∈ (𝐾 ↑_m 𝐼)) → 𝑣 ∈ 𝐼)
291	288, 290	ffvelrnd 6944	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) ∧ 𝑔 ∈ (𝐾 ↑_m 𝐼)) → (𝑔‘𝑣) ∈ 𝐾)
292	247, 71	ringidval 19654	. . . . . . . . . . . . . . 15 ⊢ (1_r‘𝑆) = (0_g‘𝑀)
293	272, 292, 248	mulg0 18622	. . . . . . . . . . . . . 14 ⊢ ((𝑔‘𝑣) ∈ 𝐾 → (0 ↑ (𝑔‘𝑣)) = (1_r‘𝑆))
294	291, 293	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) ∧ 𝑔 ∈ (𝐾 ↑_m 𝐼)) → (0 ↑ (𝑔‘𝑣)) = (1_r‘𝑆))
295	287, 294	eqtrd 2778	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) ∧ 𝑔 ∈ (𝐾 ↑_m 𝐼)) → ((𝐵‘𝑣) ↑ (𝑔‘𝑣)) = (1_r‘𝑆))
296	295	mpteq2dva 5170	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) → (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))) = (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ (1_r‘𝑆)))
297		fconstmpt 5640	. . . . . . . . . . . 12 ⊢ ((𝐾 ↑_m 𝐼) × {(1_r‘𝑆)}) = (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ (1_r‘𝑆))
298		ovexd 7290	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) → (𝐾 ↑_m 𝐼) ∈ V)
299	55, 298, 184	syl2an2r 681	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) → ((𝐾 ↑_m 𝐼) × {(1_r‘𝑆)}) = (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
300	297, 299	eqtr3id 2793	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) → (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ (1_r‘𝑆)) = (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
301	296, 300	eqtrd 2778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) → (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))) = (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
302	301, 23	suppss2 7987	. . . . . . . . 9 ⊢ (𝜑 → ((𝑣 ∈ 𝐼 ↦ (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)))) supp (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))) ⊆ (𝐵 supp 0))
303	281, 302	ssfid 8971	. . . . . . . 8 ⊢ (𝜑 → ((𝑣 ∈ 𝐼 ↦ (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)))) supp (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))) ∈ Fin)
304	275, 276, 278, 303	isfsuppd 40143	. . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ 𝐼 ↦ (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)))) finSupp (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
305	42, 41, 188, 43, 247, 56, 23, 49, 274, 304	pwsgprod 40194	. . . . . 6 ⊢ (𝜑 → ((mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) Σ_g (𝑣 ∈ 𝐼 ↦ (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))))) = (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))))))
306	198, 262, 305	3eqtrd 2782	. . . . 5 ⊢ (𝜑 → ((1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))(._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)))))) = (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))))))
307	206, 187, 306	3eqtrd 2782	. . . 4 ⊢ (𝜑 → ((𝑆 ↑_s (𝐾 ↑_m 𝐼)) Σ_g (𝑏 ∈ {𝐵} ↦ (((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)))))))) = (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))))))
308	52, 168, 307	3eqtrd 2782	. . 3 ⊢ (𝜑 → (𝑄‘𝐹) = (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))))))
309		evlsbagval.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))
310		ovexd 7290	. . 3 ⊢ (𝜑 → (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣)))) ∈ V)
311	34, 308, 309, 310	fvmptd4 40136	. 2 ⊢ (𝜑 → ((𝑄‘𝐹)‘𝐴) = (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣)))))
312	30, 311	jca 511	1 ⊢ (𝜑 → (𝐹 ∈ 𝑊 ∧ ((𝑄‘𝐹)‘𝐴) = (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣))))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 {crab 3067 Vcvv 3422 ∖ cdif 3880 ⊆ wss 3883 ifcif 4456 {csn 4558 class class class wbr 5070 ↦ cmpt 5153 × cxp 5578 ^◡ccnv 5579 ↾ cres 5582 “ cima 5583 Fun wfun 6412 Fn wfn 6413 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ∘_f cof 7509 supp csupp 7948 ↑_m cmap 8573 Fincfn 8691 finSupp cfsupp 9058 0cc0 10802 ℕcn 11903 ℕ₀cn0 12163 Basecbs 16840 ↾_s cress 16867 ._rcmulr 16889 0_gc0g 17067 Σ_g cgsu 17068 ↑_s cpws 17074 Mndcmnd 18300 ._gcmg 18615 CMndccmn 19301 mulGrpcmgp 19635 1_rcur 19652 Ringcrg 19698 CRingccrg 19699 SubRingcsubrg 19935 mPwSer cmps 21017 mPoly cmpl 21019 evalSub ces 21190

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-ofr 7512 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-fz 13169 df-fzo 13312 df-seq 13650 df-hash 13973 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-hom 16912 df-cco 16913 df-0g 17069 df-gsum 17070 df-prds 17075 df-pws 17077 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-ghm 18747 df-cntz 18838 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-srg 19657 df-ring 19700 df-cring 19701 df-rnghom 19874 df-subrg 19937 df-lmod 20040 df-lss 20109 df-lsp 20149 df-assa 20970 df-asp 20971 df-ascl 20972 df-psr 21022 df-mvr 21023 df-mpl 21024 df-evls 21192

This theorem is referenced by: mhphf 40208

W3C validator