evlsbagval - Mathbox for Steven Nguyen

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Theorem evlsbagval 40736

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 40757, 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 6857	. . . . 5 ⊢ (𝜑 → (Base‘𝑈) ∈ V)
2		evlsbagval.d	. . . . . 6 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
3		ovexd 7392	. . . . . 6 ⊢ (𝜑 → (ℕ₀ ↑_m 𝐼) ∈ V)
4	2, 3	rabexd 5290	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ V)
5		evlsbagval.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
6		evlsbagval.u	. . . . . . . . . . 11 ⊢ 𝑈 = (𝑆 ↾_s 𝑅)
7	6	subrgring 20225	. . . . . . . . . 10 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
8	5, 7	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ Ring)
9		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝑈) = (Base‘𝑈)
10		evlsbagval.o	. . . . . . . . . 10 ⊢ 1 = (1_r‘𝑈)
11	9, 10	ringidcl 19989	. . . . . . . . 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 19990	. . . . . . . . 9 ⊢ (𝑈 ∈ Ring → 0 ∈ (Base‘𝑈))
15	8, 14	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (Base‘𝑈))
16	12, 15	ifcld 4532	. . . . . . 7 ⊢ (𝜑 → if(𝑠 = 𝐵, 1 , 0 ) ∈ (Base‘𝑈))
17	16	adantr 481	. . . . . 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 7062	. . . . 5 ⊢ (𝜑 → 𝐹:𝐷⟶(Base‘𝑈))
20	1, 4, 19	elmapdd 40664	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((Base‘𝑈) ↑_m 𝐷))
21		eqid 2736	. . . . 5 ⊢ (𝐼 mPwSer 𝑈) = (𝐼 mPwSer 𝑈)
22		eqid 2736	. . . . 5 ⊢ (Base‘(𝐼 mPwSer 𝑈)) = (Base‘(𝐼 mPwSer 𝑈))
23		evlsbagval.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
24	21, 9, 2, 22, 23	psrbas 21346	. . . 4 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑈)) = ((Base‘𝑈) ↑_m 𝐷))
25	20, 24	eleqtrrd 2841	. . 3 ⊢ (𝜑 → 𝐹 ∈ (Base‘(𝐼 mPwSer 𝑈)))
26	4, 15, 18	sniffsupp 9336	. . 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 21399	. . 3 ⊢ (𝐹 ∈ 𝑊 ↔ (𝐹 ∈ (Base‘(𝐼 mPwSer 𝑈)) ∧ 𝐹 finSupp 0 ))
30	25, 26, 29	sylanbrc 583	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝑊)
31		fveq1 6841	. . . . . 6 ⊢ (𝑔 = 𝐴 → (𝑔‘𝑣) = (𝐴‘𝑣))
32	31	oveq2d 7373	. . . . 5 ⊢ (𝑔 = 𝐴 → ((𝐵‘𝑣) ↑ (𝑔‘𝑣)) = ((𝐵‘𝑣) ↑ (𝐴‘𝑣)))
33	32	mpteq2dv 5207	. . . 4 ⊢ (𝑔 = 𝐴 → (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))) = (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣))))
34	33	oveq2d 7373	. . 3 ⊢ (𝑔 = 𝐴 → (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)))) = (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣)))))
35		fveq1 6841	. . . . . . . . 9 ⊢ (𝑝 = 𝐹 → (𝑝‘𝑏) = (𝐹‘𝑏))
36	35	fveq2d 6846	. . . . . . . 8 ⊢ (𝑝 = 𝐹 → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝑝‘𝑏)) = ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝑏)))
37	36	oveq1d 7372	. . . . . . 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 5207	. . . . . 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 7373	. . . . 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 2736	. . . . . 6 ⊢ (𝑆 ↑_s (𝐾 ↑_m 𝐼)) = (𝑆 ↑_s (𝐾 ↑_m 𝐼))
43		eqid 2736	. . . . . 6 ⊢ (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
44		eqid 2736	. . . . . 6 ⊢ (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))) = (._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
45		eqid 2736	. . . . . 6 ⊢ (._r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (._r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
46		eqid 2736	. . . . . 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 2736	. . . . . 6 ⊢ (𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥})) = (𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))
48		eqid 2736	. . . . . 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 40732	. . . . 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 7392	. . . . 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 40658	. . . 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 2736	. . . . . 6 ⊢ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
54		eqid 2736	. . . . . 6 ⊢ (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
55	49	crngringd 19977	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Ring)
56		ovexd 7392	. . . . . . . 8 ⊢ (𝜑 → (𝐾 ↑_m 𝐼) ∈ V)
57	42	pwsring 20039	. . . . . . . 8 ⊢ ((𝑆 ∈ Ring ∧ (𝐾 ↑_m 𝐼) ∈ V) → (𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ Ring)
58	55, 56, 57	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ Ring)
59	58	ringcmnd 20005	. . . . . 6 ⊢ (𝜑 → (𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ CMnd)
60	58	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ Ring)
61	49	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝑅) → 𝑆 ∈ CRing)
62		ovexd 7392	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝑅) → (𝐾 ↑_m 𝐼) ∈ V)
63	41	subrgss 20223	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐾)
64	5, 63	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ⊆ 𝐾)
65	64	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑅 ⊆ 𝐾)
66	65	sselda 3944	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝑅) → 𝑥 ∈ 𝐾)
67		fconst6g 6731	. . . . . . . . . . . 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 40719	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝑅) → ((𝐾 ↑_m 𝐼) × {𝑥}) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
70	69	fmpttd 7063	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥})):𝑅⟶(Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
71		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (1_r‘𝑆) = (1_r‘𝑆)
72	6, 71	subrg1 20232	. . . . . . . . . . . . . . . 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 20230	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (1_r‘𝑆) ∈ 𝑅)
75	5, 74	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1_r‘𝑆) ∈ 𝑅)
76	73, 75	eqeltrrd 2839	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1_r‘𝑈) ∈ 𝑅)
77	10, 76	eqeltrid 2842	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ 𝑅)
78	6	subrgbas 20231	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 = (Base‘𝑈))
79	5, 78	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 = (Base‘𝑈))
80	15, 79	eleqtrrd 2841	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ 𝑅)
81	77, 80	ifcld 4532	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑠 = 𝐵, 1 , 0 ) ∈ 𝑅)
82	81	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → if(𝑠 = 𝐵, 1 , 0 ) ∈ 𝑅)
83	82, 18	fmptd 7062	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐷⟶𝑅)
84	83	ffvelcdmda 7035	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘𝑏) ∈ 𝑅)
85	70, 84	ffvelcdmd 7036	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝑏)) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
86	43, 53	mgpbas 19902	. . . . . . . . 9 ⊢ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (Base‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
87	42	pwscrng 20041	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ CRing ∧ (𝐾 ↑_m 𝐼) ∈ V) → (𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ CRing)
88	49, 56, 87	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ CRing)
89	43	crngmgp 19972	. . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∈ CMnd)
92		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ 𝐷)
93	49	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ CRing)
94		ovexd 7392	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝐼) → (𝐾 ↑_m 𝐼) ∈ V)
95		elmapi 8787	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (𝐾 ↑_m 𝐼) → 𝑎:𝐼⟶𝐾)
96	95	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → 𝑎:𝐼⟶𝐾)
97		simplr 767	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → 𝑥 ∈ 𝐼)
98	96, 97	ffvelcdmd 7036	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → (𝑎‘𝑥) ∈ 𝐾)
99	98	fmpttd 7063	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝐼) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)):(𝐾 ↑_m 𝐼)⟶𝐾)
100	42, 41, 53, 93, 94, 99	pwselbasr 40719	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐷) ∧ 𝑥 ∈ 𝐼) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
101	100	fmpttd 7063	. . . . . . . . 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 21487	. . . . . . . 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 19986	. . . . . . 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 7063	. . . . . 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 2740	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑏 → (𝑠 = 𝐵 ↔ 𝑏 = 𝐵))
106	105	ifbid 4509	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑏 → if(𝑠 = 𝐵, 1 , 0 ) = if(𝑏 = 𝐵, 1 , 0 ))
107		eldifi 4086	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ (𝐷 ∖ {𝐵}) → 𝑏 ∈ 𝐷)
108	107	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → 𝑏 ∈ 𝐷)
109	10	fvexi 6856	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
110	13	fvexi 6856	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
111	109, 110	ifex 4536	. . . . . . . . . . . . . 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 6971	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → (𝐹‘𝑏) = if(𝑏 = 𝐵, 1 , 0 ))
114		eldifsnneq 4751	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ (𝐷 ∖ {𝐵}) → ¬ 𝑏 = 𝐵)
115	114	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → ¬ 𝑏 = 𝐵)
116	115	iffalsed 4497	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → if(𝑏 = 𝐵, 1 , 0 ) = 0 )
117	113, 116	eqtrd 2776	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → (𝐹‘𝑏) = 0 )
118	117	fveq2d 6846	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝑏)) = ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘ 0 ))
119		sneq 4596	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → {𝑥} = { 0 })
120	119	xpeq2d 5663	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → ((𝐾 ↑_m 𝐼) × {𝑥}) = ((𝐾 ↑_m 𝐼) × { 0 }))
121	80	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → 0 ∈ 𝑅)
122		ovex 7390	. . . . . . . . . . . . 13 ⊢ (𝐾 ↑_m 𝐼) ∈ V
123		snex 5388	. . . . . . . . . . . . 13 ⊢ { 0 } ∈ V
124	122, 123	xpex 7687	. . . . . . . . . . . 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 6971	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘ 0 ) = ((𝐾 ↑_m 𝐼) × { 0 }))
127		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
128	6, 127	subrg0 20229	. . . . . . . . . . . . . . . 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 2794	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0_g‘𝑆) = 0 )
131	130	sneqd 4598	. . . . . . . . . . . . 13 ⊢ (𝜑 → {(0_g‘𝑆)} = { 0 })
132	131	xpeq2d 5663	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾 ↑_m 𝐼) × {(0_g‘𝑆)}) = ((𝐾 ↑_m 𝐼) × { 0 }))
133	55	ringgrpd 19973	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ Grp)
134	133	grpmndd 18760	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ Mnd)
135	42, 127	pws0g 18592	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾 ↑_m 𝐼) × {(0_g‘𝑆)}) = (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
137	132, 136	eqtr3d 2778	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐾 ↑_m 𝐼) × { 0 }) = (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
138	137	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → ((𝐾 ↑_m 𝐼) × { 0 }) = (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
139	118, 126, 138	3eqtrd 2780	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝑏)) = (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
140	139	oveq1d 7372	. . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) → (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∈ CMnd)
142	49	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ CRing)
143		ovexd 7392	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) ∧ 𝑥 ∈ 𝐼) → (𝐾 ↑_m 𝐼) ∈ V)
144	95	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) ∧ 𝑥 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → 𝑎:𝐼⟶𝐾)
145		simplr 767	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) ∧ 𝑥 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → 𝑥 ∈ 𝐼)
146	144, 145	ffvelcdmd 7036	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) ∧ 𝑥 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → (𝑎‘𝑥) ∈ 𝐾)
147	146	fmpttd 7063	. . . . . . . . . . . 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 40719	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ {𝐵})) ∧ 𝑥 ∈ 𝐼) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
149	148	fmpttd 7063	. . . . . . . . . 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 21487	. . . . . . . . 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 20011	. . . . . . . . 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 683	. . . . . . . 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 2776	. . . . . . 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 8131	. . . . . 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 7174	. . . . . . 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 6857	. . . . . . 7 ⊢ (𝜑 → (0_g‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∈ V)
157		funmpt 6539	. . . . . . . 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 8988	. . . . . . . . 9 ⊢ {𝐵} ∈ Fin
160	159	a1i 11	. . . . . . . 8 ⊢ (𝜑 → {𝐵} ∈ Fin)
161	160, 154	ssfid 9211	. . . . . . 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 40665	. . . . . 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 19690	. . . . 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 4769	. . . . . . 7 ⊢ (𝜑 → {𝐵} ⊆ 𝐷)
166	165	resmptd 5994	. . . . . 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 7373	. . . . 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 2778	. . . 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 19973	. . . . . . 7 ⊢ (𝜑 → (𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ Grp)
170	169	grpmndd 18760	. . . . . 6 ⊢ (𝜑 → (𝑆 ↑_s (𝐾 ↑_m 𝐼)) ∈ Mnd)
171		iftrue 4492	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝐵 → if(𝑠 = 𝐵, 1 , 0 ) = 1 )
172	109	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ V)
173	18, 171, 164, 172	fvmptd3 6971	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘𝐵) = 1 )
174	173	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝐵)) = ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘ 1 ))
175	10, 73	eqtr4id 2795	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 = (1_r‘𝑆))
176	175	fveq2d 6846	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘ 1 ) = ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(1_r‘𝑆)))
177		sneq 4596	. . . . . . . . . . . . 13 ⊢ (𝑥 = (1_r‘𝑆) → {𝑥} = {(1_r‘𝑆)})
178	177	xpeq2d 5663	. . . . . . . . . . . 12 ⊢ (𝑥 = (1_r‘𝑆) → ((𝐾 ↑_m 𝐼) × {𝑥}) = ((𝐾 ↑_m 𝐼) × {(1_r‘𝑆)}))
179		snex 5388	. . . . . . . . . . . . . 14 ⊢ {(1_r‘𝑆)} ∈ V
180	179	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → {(1_r‘𝑆)} ∈ V)
181	56, 180	xpexd 7685	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐾 ↑_m 𝐼) × {(1_r‘𝑆)}) ∈ V)
182	47, 178, 75, 181	fvmptd3 6971	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(1_r‘𝑆)) = ((𝐾 ↑_m 𝐼) × {(1_r‘𝑆)}))
183	176, 182	eqtrd 2776	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘ 1 ) = ((𝐾 ↑_m 𝐼) × {(1_r‘𝑆)}))
184	42, 71	pws1 20040	. . . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐾 ↑_m 𝐼) × {(1_r‘𝑆)}) = (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
186	174, 183, 185	3eqtrd 2780	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝐵)) = (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
187	186	oveq1d 7372	. . . . . . . 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 2736	. . . . . . . . 9 ⊢ (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) = (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))
189	49	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ CRing)
190		ovexd 7392	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐾 ↑_m 𝐼) ∈ V)
191	95	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → 𝑎:𝐼⟶𝐾)
192		simplr 767	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → 𝑥 ∈ 𝐼)
193	191, 192	ffvelcdmd 7036	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → (𝑎‘𝑥) ∈ 𝐾)
194	193	fmpttd 7063	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)):(𝐾 ↑_m 𝐼)⟶𝐾)
195	42, 41, 53, 189, 190, 194	pwselbasr 40719	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
196	195	fmpttd 7063	. . . . . . . . . 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 21487	. . . . . . . . 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 40691	. . . . . . . 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 2776	. . . . . . 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 2838	. . . . . 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 6847	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝑏)) = ((𝑥 ∈ 𝑅 ↦ ((𝐾 ↑_m 𝐼) × {𝑥}))‘(𝐹‘𝐵)))
202		oveq1 7364	. . . . . . . . 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 7373	. . . . . . . 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 7375	. . . . . . 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 19731	. . . . . 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 1371	. . . . 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 21320	. . . . . . . . . . 11 ⊢ (𝐵 ∈ 𝐷 → 𝐵:𝐼⟶ℕ₀)
208	164, 207	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵:𝐼⟶ℕ₀)
209	208	ffnd 6669	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 Fn 𝐼)
210	122	mptex 7173	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)) ∈ V
211	210, 48	fnmpti 6644	. . . . . . . . . 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 4178	. . . . . . . . 9 ⊢ (𝐼 ∩ 𝐼) = 𝐼
214		eqidd 2737	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → (𝐵‘𝑣) = (𝐵‘𝑣))
215		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑣 → (𝑎‘𝑥) = (𝑎‘𝑣))
216	215	mpteq2dv 5207	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑥)) = (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣)))
217		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → 𝑣 ∈ 𝐼)
218	122	mptex 7173	. . . . . . . . . . 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 6971	. . . . . . . . 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 7626	. . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → 𝑆 ∈ CRing)
223		ovexd 7392	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → (𝐾 ↑_m 𝐼) ∈ V)
224	43	ringmgp 19970	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → (mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∈ Mnd)
227	208	ffvelcdmda 7035	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → (𝐵‘𝑣) ∈ ℕ₀)
228	95	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → 𝑎:𝐼⟶𝐾)
229		simplr 767	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → 𝑣 ∈ 𝐼)
230	228, 229	ffvelcdmd 7036	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑎 ∈ (𝐾 ↑_m 𝐼)) → (𝑎‘𝑣) ∈ 𝐾)
231	230	fmpttd 7063	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣)):(𝐾 ↑_m 𝐼)⟶𝐾)
232	42, 41, 53, 222, 223, 231	pwselbasr 40719	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣)) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
233	86, 44, 226, 227, 232	mulgnn0cld 18897	. . . . . . . . . . . 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))))
234	42, 41, 53, 222, 223, 233	pwselbas 17371	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → ((𝐵‘𝑣)(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣))):(𝐾 ↑_m 𝐼)⟶𝐾)
235	234	ffnd 6669	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → ((𝐵‘𝑣)(._g‘(mulGrp‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))(𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣))) Fn (𝐾 ↑_m 𝐼))
236		ovex 7390	. . . . . . . . . . . 12 ⊢ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)) ∈ V
237		eqid 2736	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))) = (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)))
238	236, 237	fnmpti 6644	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))) Fn (𝐾 ↑_m 𝐼)
239	238	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝐼) → (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))) Fn (𝐾 ↑_m 𝐼))
240		eqid 2736	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣)) = (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣))
241		fveq1 6841	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑙 → (𝑎‘𝑣) = (𝑙‘𝑣))
242		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → 𝑙 ∈ (𝐾 ↑_m 𝐼))
243		fvexd 6857	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → (𝑙‘𝑣) ∈ V)
244	240, 241, 242, 243	fvmptd3 6971	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → ((𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣))‘𝑙) = (𝑙‘𝑣))
245	244	oveq2d 7373	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → ((𝐵‘𝑣) ↑ ((𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣))‘𝑙)) = ((𝐵‘𝑣) ↑ (𝑙‘𝑣)))
246		evlsbagval.m	. . . . . . . . . . . 12 ⊢ 𝑀 = (mulGrp‘𝑆)
247		evlsbagval.e	. . . . . . . . . . . 12 ⊢ ↑ = (._g‘𝑀)
248	55	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → 𝑆 ∈ Ring)
249		ovexd 7392	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → (𝐾 ↑_m 𝐼) ∈ V)
250	227	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → (𝐵‘𝑣) ∈ ℕ₀)
251	232	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → (𝑎 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑎‘𝑣)) ∈ (Base‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
252	42, 53, 43, 246, 44, 247, 248, 249, 250, 251, 242	pwsexpg 20044	. . . . . . . . . . 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)))
253		fveq1 6841	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑙 → (𝑔‘𝑣) = (𝑙‘𝑣))
254	253	oveq2d 7373	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑙 → ((𝐵‘𝑣) ↑ (𝑔‘𝑣)) = ((𝐵‘𝑣) ↑ (𝑙‘𝑣)))
255		ovexd 7392	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → ((𝐵‘𝑣) ↑ (𝑙‘𝑣)) ∈ V)
256	237, 254, 242, 255	fvmptd3 6971	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝐼) ∧ 𝑙 ∈ (𝐾 ↑_m 𝐼)) → ((𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)))‘𝑙) = ((𝐵‘𝑣) ↑ (𝑙‘𝑣)))
257	245, 252, 256	3eqtr4d 2786	. . . . . . . . . 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))
258	235, 239, 257	eqfnfvd 6985	. . . . . . . . 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))))
259	258	mpteq2dva 5205	. . . . . . . 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)))))
260	221, 259	eqtrd 2776	. . . . . . 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)))))
261	260	oveq2d 7373	. . . . . 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))))))
262	246, 41	mgpbas 19902	. . . . . . . 8 ⊢ 𝐾 = (Base‘𝑀)
263	246	crngmgp 19972	. . . . . . . . . . 11 ⊢ (𝑆 ∈ CRing → 𝑀 ∈ CMnd)
264	49, 263	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ CMnd)
265	264	cmnmndd 19586	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ Mnd)
266	265	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑣 ∈ 𝐼)) → 𝑀 ∈ Mnd)
267	227	adantrl 714	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑣 ∈ 𝐼)) → (𝐵‘𝑣) ∈ ℕ₀)
268		elmapi 8787	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝐾 ↑_m 𝐼) → 𝑔:𝐼⟶𝐾)
269	268	ad2antrl 726	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑣 ∈ 𝐼)) → 𝑔:𝐼⟶𝐾)
270		simprr 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑣 ∈ 𝐼)) → 𝑣 ∈ 𝐼)
271	269, 270	ffvelcdmd 7036	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑣 ∈ 𝐼)) → (𝑔‘𝑣) ∈ 𝐾)
272	262, 247, 266, 267, 271	mulgnn0cld 18897	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ (𝐾 ↑_m 𝐼) ∧ 𝑣 ∈ 𝐼)) → ((𝐵‘𝑣) ↑ (𝑔‘𝑣)) ∈ 𝐾)
273	23	mptexd 7174	. . . . . . . 8 ⊢ (𝜑 → (𝑣 ∈ 𝐼 ↦ (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)))) ∈ V)
274		fvexd 6857	. . . . . . . 8 ⊢ (𝜑 → (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))) ∈ V)
275		funmpt 6539	. . . . . . . . 9 ⊢ Fun (𝑣 ∈ 𝐼 ↦ (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))))
276	275	a1i 11	. . . . . . . 8 ⊢ (𝜑 → Fun (𝑣 ∈ 𝐼 ↦ (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)))))
277	2	psrbagfsupp 21322	. . . . . . . . . . 11 ⊢ (𝐵 ∈ 𝐷 → 𝐵 finSupp 0)
278	164, 277	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 finSupp 0)
279	278	fsuppimpd 9312	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 supp 0) ∈ Fin)
280		ssidd 3967	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 supp 0) ⊆ (𝐵 supp 0))
281		c0ex 11149	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ V
282	281	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ V)
283	208, 280, 23, 282	suppssr 8127	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) → (𝐵‘𝑣) = 0)
284	283	oveq1d 7372	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) → ((𝐵‘𝑣) ↑ (𝑔‘𝑣)) = (0 ↑ (𝑔‘𝑣)))
285	284	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) ∧ 𝑔 ∈ (𝐾 ↑_m 𝐼)) → ((𝐵‘𝑣) ↑ (𝑔‘𝑣)) = (0 ↑ (𝑔‘𝑣)))
286	268	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) ∧ 𝑔 ∈ (𝐾 ↑_m 𝐼)) → 𝑔:𝐼⟶𝐾)
287		eldifi 4086	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ (𝐼 ∖ (𝐵 supp 0)) → 𝑣 ∈ 𝐼)
288	287	ad2antlr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) ∧ 𝑔 ∈ (𝐾 ↑_m 𝐼)) → 𝑣 ∈ 𝐼)
289	286, 288	ffvelcdmd 7036	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) ∧ 𝑔 ∈ (𝐾 ↑_m 𝐼)) → (𝑔‘𝑣) ∈ 𝐾)
290	246, 71	ringidval 19915	. . . . . . . . . . . . . . 15 ⊢ (1_r‘𝑆) = (0_g‘𝑀)
291	262, 290, 247	mulg0 18879	. . . . . . . . . . . . . 14 ⊢ ((𝑔‘𝑣) ∈ 𝐾 → (0 ↑ (𝑔‘𝑣)) = (1_r‘𝑆))
292	289, 291	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) ∧ 𝑔 ∈ (𝐾 ↑_m 𝐼)) → (0 ↑ (𝑔‘𝑣)) = (1_r‘𝑆))
293	285, 292	eqtrd 2776	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) ∧ 𝑔 ∈ (𝐾 ↑_m 𝐼)) → ((𝐵‘𝑣) ↑ (𝑔‘𝑣)) = (1_r‘𝑆))
294	293	mpteq2dva 5205	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) → (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))) = (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ (1_r‘𝑆)))
295		fconstmpt 5694	. . . . . . . . . . . 12 ⊢ ((𝐾 ↑_m 𝐼) × {(1_r‘𝑆)}) = (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ (1_r‘𝑆))
296		ovexd 7392	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) → (𝐾 ↑_m 𝐼) ∈ V)
297	55, 296, 184	syl2an2r 683	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) → ((𝐾 ↑_m 𝐼) × {(1_r‘𝑆)}) = (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
298	295, 297	eqtr3id 2790	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) → (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ (1_r‘𝑆)) = (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
299	294, 298	eqtrd 2776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑣 ∈ (𝐼 ∖ (𝐵 supp 0))) → (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))) = (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
300	299, 23	suppss2 8131	. . . . . . . . 9 ⊢ (𝜑 → ((𝑣 ∈ 𝐼 ↦ (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)))) supp (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))) ⊆ (𝐵 supp 0))
301	279, 300	ssfid 9211	. . . . . . . 8 ⊢ (𝜑 → ((𝑣 ∈ 𝐼 ↦ (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)))) supp (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼)))) ∈ Fin)
302	273, 274, 276, 301	isfsuppd 40665	. . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ 𝐼 ↦ (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣)))) finSupp (1_r‘(𝑆 ↑_s (𝐾 ↑_m 𝐼))))
303	42, 41, 188, 43, 246, 56, 23, 49, 272, 302	pwsgprod 40720	. . . . . 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))))))
304	198, 261, 303	3eqtrd 2780	. . . . 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))))))
305	206, 187, 304	3eqtrd 2780	. . . 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))))))
306	52, 168, 305	3eqtrd 2780	. . 3 ⊢ (𝜑 → (𝑄‘𝐹) = (𝑔 ∈ (𝐾 ↑_m 𝐼) ↦ (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝑔‘𝑣))))))
307		evlsbagval.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))
308		ovexd 7392	. . 3 ⊢ (𝜑 → (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣)))) ∈ V)
309	34, 306, 307, 308	fvmptd4 40658	. 2 ⊢ (𝜑 → ((𝑄‘𝐹)‘𝐴) = (𝑀 Σ_g (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣)))))
310	30, 309	jca 512	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 396 = wceq 1541 ∈ wcel 2106 {crab 3407 Vcvv 3445 ∖ cdif 3907 ⊆ wss 3910 ifcif 4486 {csn 4586 class class class wbr 5105 ↦ cmpt 5188 × cxp 5631 ^◡ccnv 5632 ↾ cres 5635 “ cima 5636 Fun wfun 6490 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ∘_f cof 7615 supp csupp 8092 ↑_m cmap 8765 Fincfn 8883 finSupp cfsupp 9305 0cc0 11051 ℕcn 12153 ℕ₀cn0 12413 Basecbs 17083 ↾_s cress 17112 ._rcmulr 17134 0_gc0g 17321 Σ_g cgsu 17322 ↑_s cpws 17328 Mndcmnd 18556 ._gcmg 18872 CMndccmn 19562 mulGrpcmgp 19896 1_rcur 19913 Ringcrg 19964 CRingccrg 19965 SubRingcsubrg 20218 mPwSer cmps 21306 mPoly cmpl 21308 evalSub ces 21480

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-ofr 7618 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-fz 13425 df-fzo 13568 df-seq 13907 df-hash 14231 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-hom 17157 df-cco 17158 df-0g 17323 df-gsum 17324 df-prds 17329 df-pws 17331 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-ghm 19006 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-srg 19918 df-ring 19966 df-cring 19967 df-rnghom 20146 df-subrg 20220 df-lmod 20324 df-lss 20393 df-lsp 20433 df-assa 21259 df-asp 21260 df-ascl 21261 df-psr 21311 df-mvr 21312 df-mpl 21313 df-evls 21482

This theorem is referenced by: mhphf 40757

W3C validator