Theorem psrmonprod 33851

Description: Finite product of bags of variables in a power series. Here the function 𝐺 maps a bag of variables to the corresponding monomial. (Contributed by Thierry Arnoux, 16-Mar-2026.)

Hypotheses

Ref	Expression
psrmonprod.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
psrmonprod.b	⊢ 𝐵 = (Base‘𝑆)
psrmonprod.r	⊢ (𝜑 → 𝑅 ∈ CRing)
psrmonprod.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
psrmonprod.d	⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
psrmonprod.a	⊢ (𝜑 → 𝐴 ∈ Fin)
psrmonprod.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐷)
psrmonprod.1	⊢ 1 = (1r‘𝑅)
psrmonprod.0	⊢ 0 = (0g‘𝑅)
psrmonprod.m	⊢ 𝑀 = (mulGrp‘𝑆)
psrmonprod.g	⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 = 𝑦, 1 , 0 )))

Assertion

Ref	Expression
psrmonprod	⊢ (𝜑 → (𝑀 Σg (𝐺 ∘ 𝐹)) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)‘𝑖))))))

Distinct variable groups:   0 ,ℎ,𝑦,𝑧   𝑦, 1 ,𝑧   𝐴,𝑖,𝑥,𝑦,𝑧   𝑦,𝐵,𝑧   𝐷,𝑖,𝑦,𝑧   ℎ,𝐹,𝑖,𝑥,𝑦,𝑧   𝑦,𝐺,𝑧   ℎ,𝐼,𝑦,𝑧,𝑖,𝑥   𝑅,ℎ,𝑦,𝑧   𝑦,𝑆,𝑧   𝑦,𝑉,𝑧   𝜑,𝑖,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(ℎ)   𝐴(ℎ)   𝐵(𝑥,ℎ,𝑖)   𝐷(𝑥,ℎ)   𝑅(𝑥,𝑖)   𝑆(𝑥,ℎ,𝑖)   1 (𝑥,ℎ,𝑖)   𝐺(𝑥,ℎ,𝑖)   𝑀(𝑥,𝑦,𝑧,ℎ,𝑖)   𝑉(𝑥,ℎ,𝑖)   0 (𝑥,𝑖)

Proof of Theorem psrmonprod

Dummy variables 𝑎 𝑏 𝑓 𝑗 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psrmonprod.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐷)
2	1	ffvelcdmda 7065	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ 𝐷)
3	1	feqmptd 6935	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝑘)))
4		fvexd 6882	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → (Base‘𝑅) ∈ V)
5		psrmonprod.d	. . . . . . . . . 10 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
6		ovex 7429	. . . . . . . . . 10 ⊢ (ℕ0 ↑m 𝐼) ∈ V
7	5, 6	rabex2 5298	. . . . . . . . 9 ⊢ 𝐷 ∈ V
8	7	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝐷 ∈ V)
9		eqid 2763	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
10		psrmonprod.1	. . . . . . . . . . . 12 ⊢ 1 = (1r‘𝑅)
11		psrmonprod.r	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ CRing)
12	11	crngringd 20306	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ Ring)
13	9, 10, 12	ringidcld 20326	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ (Base‘𝑅))
14	13	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐷) ∧ 𝑧 ∈ 𝐷) → 1 ∈ (Base‘𝑅))
15	11	crnggrpd 20307	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ Grp)
16		psrmonprod.0	. . . . . . . . . . . . 13 ⊢ 0 = (0g‘𝑅)
17	9, 16	grpidcl 19017	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Grp → 0 ∈ (Base‘𝑅))
18	15, 17	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ (Base‘𝑅))
19	18	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐷) ∧ 𝑧 ∈ 𝐷) → 0 ∈ (Base‘𝑅))
20	14, 19	ifcld 4528	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐷) ∧ 𝑧 ∈ 𝐷) → if(𝑧 = 𝑦, 1 , 0 ) ∈ (Base‘𝑅))
21	20	fmpttd 7096	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → (𝑧 ∈ 𝐷 ↦ if(𝑧 = 𝑦, 1 , 0 )):𝐷⟶(Base‘𝑅))
22	4, 8, 21	elmapdd 8822	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → (𝑧 ∈ 𝐷 ↦ if(𝑧 = 𝑦, 1 , 0 )) ∈ ((Base‘𝑅) ↑m 𝐷))
23		psrmonprod.s	. . . . . . . . 9 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
24	5	psrbasfsupp 33810	. . . . . . . . 9 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
25		psrmonprod.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑆)
26		psrmonprod.i	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
27	23, 9, 24, 25, 26	psrbas 21993	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = ((Base‘𝑅) ↑m 𝐷))
28	27	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝐵 = ((Base‘𝑅) ↑m 𝐷))
29	22, 28	eleqtrrd 2866	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → (𝑧 ∈ 𝐷 ↦ if(𝑧 = 𝑦, 1 , 0 )) ∈ 𝐵)
30		psrmonprod.g	. . . . . 6 ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 = 𝑦, 1 , 0 )))
31	29, 30	fmptd 7095	. . . . 5 ⊢ (𝜑 → 𝐺:𝐷⟶𝐵)
32	31	feqmptd 6935	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐷 ↦ (𝐺‘𝑦)))
33		fveq2 6867	. . . 4 ⊢ (𝑦 = (𝐹‘𝑘) → (𝐺‘𝑦) = (𝐺‘(𝐹‘𝑘)))
34	2, 3, 32, 33	fmptco 7111	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑘 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑘))))
35	34	oveq2d 7412	. 2 ⊢ (𝜑 → (𝑀 Σg (𝐺 ∘ 𝐹)) = (𝑀 Σg (𝑘 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑘)))))
36		mpteq1 5190	. . . . 5 ⊢ (𝑎 = ∅ → (𝑘 ∈ 𝑎 ↦ (𝐺‘(𝐹‘𝑘))) = (𝑘 ∈ ∅ ↦ (𝐺‘(𝐹‘𝑘))))
37	36	oveq2d 7412	. . . 4 ⊢ (𝑎 = ∅ → (𝑀 Σg (𝑘 ∈ 𝑎 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝑀 Σg (𝑘 ∈ ∅ ↦ (𝐺‘(𝐹‘𝑘)))))
38		mpteq1 5190	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑥 ∈ 𝑎 ↦ ((𝐹‘𝑥)‘𝑖)) = (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖)))
39	38	oveq2d 7412	. . . . . 6 ⊢ (𝑎 = ∅ → (ℂfld Σg (𝑥 ∈ 𝑎 ↦ ((𝐹‘𝑥)‘𝑖))) = (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖))))
40	39	mpteq2dv 5195	. . . . 5 ⊢ (𝑎 = ∅ → (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑎 ↦ ((𝐹‘𝑥)‘𝑖)))) = (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖)))))
41	40	fveq2d 6871	. . . 4 ⊢ (𝑎 = ∅ → (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑎 ↦ ((𝐹‘𝑥)‘𝑖))))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖))))))
42	37, 41	eqeq12d 2779	. . 3 ⊢ (𝑎 = ∅ → ((𝑀 Σg (𝑘 ∈ 𝑎 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑎 ↦ ((𝐹‘𝑥)‘𝑖))))) ↔ (𝑀 Σg (𝑘 ∈ ∅ ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖)))))))
43		mpteq1 5190	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝑘 ∈ 𝑎 ↦ (𝐺‘(𝐹‘𝑘))) = (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘))))
44	43	oveq2d 7412	. . . 4 ⊢ (𝑎 = 𝑏 → (𝑀 Σg (𝑘 ∈ 𝑎 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))))
45		mpteq1 5190	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑥 ∈ 𝑎 ↦ ((𝐹‘𝑥)‘𝑖)) = (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))
46	45	oveq2d 7412	. . . . . 6 ⊢ (𝑎 = 𝑏 → (ℂfld Σg (𝑥 ∈ 𝑎 ↦ ((𝐹‘𝑥)‘𝑖))) = (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖))))
47	46	mpteq2dv 5195	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑎 ↦ ((𝐹‘𝑥)‘𝑖)))) = (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))
48	47	fveq2d 6871	. . . 4 ⊢ (𝑎 = 𝑏 → (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑎 ↦ ((𝐹‘𝑥)‘𝑖))))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖))))))
49	44, 48	eqeq12d 2779	. . 3 ⊢ (𝑎 = 𝑏 → ((𝑀 Σg (𝑘 ∈ 𝑎 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑎 ↦ ((𝐹‘𝑥)‘𝑖))))) ↔ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))))
50		mpteq1 5190	. . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑓}) → (𝑘 ∈ 𝑎 ↦ (𝐺‘(𝐹‘𝑘))) = (𝑘 ∈ (𝑏 ∪ {𝑓}) ↦ (𝐺‘(𝐹‘𝑘))))
51	50	oveq2d 7412	. . . 4 ⊢ (𝑎 = (𝑏 ∪ {𝑓}) → (𝑀 Σg (𝑘 ∈ 𝑎 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝑀 Σg (𝑘 ∈ (𝑏 ∪ {𝑓}) ↦ (𝐺‘(𝐹‘𝑘)))))
52		mpteq1 5190	. . . . . . 7 ⊢ (𝑎 = (𝑏 ∪ {𝑓}) → (𝑥 ∈ 𝑎 ↦ ((𝐹‘𝑥)‘𝑖)) = (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖)))
53	52	oveq2d 7412	. . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑓}) → (ℂfld Σg (𝑥 ∈ 𝑎 ↦ ((𝐹‘𝑥)‘𝑖))) = (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖))))
54	53	mpteq2dv 5195	. . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑓}) → (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑎 ↦ ((𝐹‘𝑥)‘𝑖)))) = (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖)))))
55	54	fveq2d 6871	. . . 4 ⊢ (𝑎 = (𝑏 ∪ {𝑓}) → (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑎 ↦ ((𝐹‘𝑥)‘𝑖))))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖))))))
56	51, 55	eqeq12d 2779	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑓}) → ((𝑀 Σg (𝑘 ∈ 𝑎 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑎 ↦ ((𝐹‘𝑥)‘𝑖))))) ↔ (𝑀 Σg (𝑘 ∈ (𝑏 ∪ {𝑓}) ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖)))))))
57		mpteq1 5190	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑘 ∈ 𝑎 ↦ (𝐺‘(𝐹‘𝑘))) = (𝑘 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑘))))
58	57	oveq2d 7412	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑀 Σg (𝑘 ∈ 𝑎 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝑀 Σg (𝑘 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑘)))))
59		mpteq1 5190	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑥 ∈ 𝑎 ↦ ((𝐹‘𝑥)‘𝑖)) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)‘𝑖)))
60	59	oveq2d 7412	. . . . . 6 ⊢ (𝑎 = 𝐴 → (ℂfld Σg (𝑥 ∈ 𝑎 ↦ ((𝐹‘𝑥)‘𝑖))) = (ℂfld Σg (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)‘𝑖))))
61	60	mpteq2dv 5195	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑎 ↦ ((𝐹‘𝑥)‘𝑖)))) = (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)‘𝑖)))))
62	61	fveq2d 6871	. . . 4 ⊢ (𝑎 = 𝐴 → (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑎 ↦ ((𝐹‘𝑥)‘𝑖))))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)‘𝑖))))))
63	58, 62	eqeq12d 2779	. . 3 ⊢ (𝑎 = 𝐴 → ((𝑀 Σg (𝑘 ∈ 𝑎 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑎 ↦ ((𝐹‘𝑥)‘𝑖))))) ↔ (𝑀 Σg (𝑘 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)‘𝑖)))))))
64		psrmonprod.m	. . . . . 6 ⊢ 𝑀 = (mulGrp‘𝑆)
65		eqid 2763	. . . . . 6 ⊢ (1r‘𝑆) = (1r‘𝑆)
66	64, 65	ringidval 20243	. . . . 5 ⊢ (1r‘𝑆) = (0g‘𝑀)
67	66	gsum0 18728	. . . 4 ⊢ (𝑀 Σg ∅) = (1r‘𝑆)
68		mpt0 6663	. . . . . 6 ⊢ (𝑘 ∈ ∅ ↦ (𝐺‘(𝐹‘𝑘))) = ∅
69	68	oveq2i 7407	. . . . 5 ⊢ (𝑀 Σg (𝑘 ∈ ∅ ↦ (𝐺‘(𝐹‘𝑘)))) = (𝑀 Σg ∅)
70	69	a1i 11	. . . 4 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ ∅ ↦ (𝐺‘(𝐹‘𝑘)))) = (𝑀 Σg ∅))
71		mpt0 6663	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖)) = ∅
72	71	oveq2i 7407	. . . . . . . . . . . . . . 15 ⊢ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖))) = (ℂfld Σg ∅)
73		cnfld0 21455	. . . . . . . . . . . . . . . 16 ⊢ 0 = (0g‘ℂfld)
74	73	gsum0 18728	. . . . . . . . . . . . . . 15 ⊢ (ℂfld Σg ∅) = 0
75	72, 74	eqtri 2786	. . . . . . . . . . . . . 14 ⊢ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖))) = 0
76	75	mpteq2i 5197	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖)))) = (𝑖 ∈ 𝐼 ↦ 0)
77		fconstmpt 5710	. . . . . . . . . . . . 13 ⊢ (𝐼 × {0}) = (𝑖 ∈ 𝐼 ↦ 0)
78	76, 77	eqtr4i 2789	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖)))) = (𝐼 × {0})
79	78	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖)))) = (𝐼 × {0}))
80	79	eqeq2d 2774	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 = (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖)))) ↔ 𝑦 = (𝐼 × {0})))
81	80	biimpa 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 = (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖))))) → 𝑦 = (𝐼 × {0}))
82	81	eqeq2d 2774	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 = (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖))))) → (𝑧 = 𝑦 ↔ 𝑧 = (𝐼 × {0})))
83	82	ifbid 4505	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 = (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖))))) → if(𝑧 = 𝑦, 1 , 0 ) = if(𝑧 = (𝐼 × {0}), 1 , 0 ))
84	83	mpteq2dv 5195	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖))))) → (𝑧 ∈ 𝐷 ↦ if(𝑧 = 𝑦, 1 , 0 )) = (𝑧 ∈ 𝐷 ↦ if(𝑧 = (𝐼 × {0}), 1 , 0 )))
85	23, 26, 12, 24, 16, 10, 65	psr1 22029	. . . . . . 7 ⊢ (𝜑 → (1r‘𝑆) = (𝑧 ∈ 𝐷 ↦ if(𝑧 = (𝐼 × {0}), 1 , 0 )))
86	85	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖))))) → (1r‘𝑆) = (𝑧 ∈ 𝐷 ↦ if(𝑧 = (𝐼 × {0}), 1 , 0 )))
87	84, 86	eqtr4d 2801	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖))))) → (𝑧 ∈ 𝐷 ↦ if(𝑧 = 𝑦, 1 , 0 )) = (1r‘𝑆))
88		breq1 5104	. . . . . . 7 ⊢ (ℎ = (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖)))) → (ℎ finSupp 0 ↔ (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖)))) finSupp 0))
89		nn0ex 12497	. . . . . . . . . 10 ⊢ ℕ0 ∈ V
90	89	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℕ0 ∈ V)
91		0nn0 12506	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ0
92	91	fconst6 6754	. . . . . . . . . 10 ⊢ (𝐼 × {0}):𝐼⟶ℕ0
93	92	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (𝐼 × {0}):𝐼⟶ℕ0)
94	90, 26, 93	elmapdd 8822	. . . . . . . 8 ⊢ (𝜑 → (𝐼 × {0}) ∈ (ℕ0 ↑m 𝐼))
95	78, 94	eqeltrid 2867	. . . . . . 7 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖)))) ∈ (ℕ0 ↑m 𝐼))
96	91	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℕ0)
97	26, 96	fczfsuppd 9330	. . . . . . . 8 ⊢ (𝜑 → (𝐼 × {0}) finSupp 0)
98	78, 97	eqbrtrid 5136	. . . . . . 7 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖)))) finSupp 0)
99	88, 95, 98	elrabd 3653	. . . . . 6 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖)))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
100	99, 5	eleqtrrdi 2874	. . . . 5 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖)))) ∈ 𝐷)
101		fvexd 6882	. . . . 5 ⊢ (𝜑 → (1r‘𝑆) ∈ V)
102	30, 87, 100, 101	fvmptd2 6984	. . . 4 ⊢ (𝜑 → (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖))))) = (1r‘𝑆))
103	67, 70, 102	3eqtr4a 2824	. . 3 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ ∅ ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ ∅ ↦ ((𝐹‘𝑥)‘𝑖))))))
104		2fveq3 6872	. . . . . . . 8 ⊢ (𝑘 = 𝑙 → (𝐺‘(𝐹‘𝑘)) = (𝐺‘(𝐹‘𝑙)))
105	104	cbvmptv 5205	. . . . . . 7 ⊢ (𝑘 ∈ (𝑏 ∪ {𝑓}) ↦ (𝐺‘(𝐹‘𝑘))) = (𝑙 ∈ (𝑏 ∪ {𝑓}) ↦ (𝐺‘(𝐹‘𝑙)))
106	105	oveq2i 7407	. . . . . 6 ⊢ (𝑀 Σg (𝑘 ∈ (𝑏 ∪ {𝑓}) ↦ (𝐺‘(𝐹‘𝑘)))) = (𝑀 Σg (𝑙 ∈ (𝑏 ∪ {𝑓}) ↦ (𝐺‘(𝐹‘𝑙))))
107	64, 25	mgpbas 20201	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑀)
108		eqid 2763	. . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆)
109	64, 108	mgpplusg 20200	. . . . . . . 8 ⊢ (.r‘𝑆) = (+g‘𝑀)
110	23, 26, 11	psrcrng 22030	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ CRing)
111	64	crngmgp 20301	. . . . . . . . . 10 ⊢ (𝑆 ∈ CRing → 𝑀 ∈ CMnd)
112	110, 111	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ CMnd)
113	112	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))) → 𝑀 ∈ CMnd)
114		psrmonprod.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ Fin)
115	114	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝐴 ∈ Fin)
116		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑏 ⊆ 𝐴)
117	115, 116	ssfid 9213	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑏 ∈ Fin)
118	117	ad2antrr 736	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))) → 𝑏 ∈ Fin)
119	31	ad4antr 742	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))) ∧ 𝑙 ∈ 𝑏) → 𝐺:𝐷⟶𝐵)
120	1	ad4antr 742	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))) ∧ 𝑙 ∈ 𝑏) → 𝐹:𝐴⟶𝐷)
121		simpllr 785	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))) → 𝑏 ⊆ 𝐴)
122	121	sselda 3937	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))) ∧ 𝑙 ∈ 𝑏) → 𝑙 ∈ 𝐴)
123	120, 122	ffvelcdmd 7066	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))) ∧ 𝑙 ∈ 𝑏) → (𝐹‘𝑙) ∈ 𝐷)
124	119, 123	ffvelcdmd 7066	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))) ∧ 𝑙 ∈ 𝑏) → (𝐺‘(𝐹‘𝑙)) ∈ 𝐵)
125		simplr 778	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))) → 𝑓 ∈ (𝐴 ∖ 𝑏))
126	125	eldifbd 3918	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))) → ¬ 𝑓 ∈ 𝑏)
127	31	ad3antrrr 740	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))) → 𝐺:𝐷⟶𝐵)
128	1	ad3antrrr 740	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))) → 𝐹:𝐴⟶𝐷)
129	125	eldifad 3917	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))) → 𝑓 ∈ 𝐴)
130	128, 129	ffvelcdmd 7066	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))) → (𝐹‘𝑓) ∈ 𝐷)
131	127, 130	ffvelcdmd 7066	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))) → (𝐺‘(𝐹‘𝑓)) ∈ 𝐵)
132		2fveq3 6872	. . . . . . . 8 ⊢ (𝑙 = 𝑓 → (𝐺‘(𝐹‘𝑙)) = (𝐺‘(𝐹‘𝑓)))
133	107, 109, 113, 118, 124, 125, 126, 131, 132	gsumunsn 20010	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))) → (𝑀 Σg (𝑙 ∈ (𝑏 ∪ {𝑓}) ↦ (𝐺‘(𝐹‘𝑙)))) = ((𝑀 Σg (𝑙 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑙))))(.r‘𝑆)(𝐺‘(𝐹‘𝑓))))
134	104	cbvmptv 5205	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘))) = (𝑙 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑙)))
135	134	oveq2i 7407	. . . . . . . . . 10 ⊢ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝑀 Σg (𝑙 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑙))))
136		id 22	. . . . . . . . . 10 ⊢ ((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖))))) → (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖))))))
137	135, 136	eqtr3id 2812	. . . . . . . . 9 ⊢ ((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖))))) → (𝑀 Σg (𝑙 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑙)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖))))))
138	137	oveq1d 7411	. . . . . . . 8 ⊢ ((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖))))) → ((𝑀 Σg (𝑙 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑙))))(.r‘𝑆)(𝐺‘(𝐹‘𝑓))) = ((𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))(.r‘𝑆)(𝐺‘(𝐹‘𝑓))))
139	138	adantl 485	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))) → ((𝑀 Σg (𝑙 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑙))))(.r‘𝑆)(𝐺‘(𝐹‘𝑓))) = ((𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))(.r‘𝑆)(𝐺‘(𝐹‘𝑓))))
140	26	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → 𝐼 ∈ 𝑉)
141	12	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → 𝑅 ∈ Ring)
142		breq1 5104	. . . . . . . . . . . 12 ⊢ (ℎ = (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))) → (ℎ finSupp 0 ↔ (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))) finSupp 0))
143	89	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → ℕ0 ∈ V)
144		cnfldfld 33531	. . . . . . . . . . . . . . . . 17 ⊢ ℂfld ∈ Field
145		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℂfld ∈ Field → ℂfld ∈ Field)
146	145	fldcrngd 20801	. . . . . . . . . . . . . . . . . 18 ⊢ (ℂfld ∈ Field → ℂfld ∈ CRing)
147		crngring 20305	. . . . . . . . . . . . . . . . . 18 ⊢ (ℂfld ∈ CRing → ℂfld ∈ Ring)
148		ringcmn 20342	. . . . . . . . . . . . . . . . . 18 ⊢ (ℂfld ∈ Ring → ℂfld ∈ CMnd)
149	146, 147, 148	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (ℂfld ∈ Field → ℂfld ∈ CMnd)
150	144, 149	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ℂfld ∈ CMnd
151	150	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑖 ∈ 𝐼) → ℂfld ∈ CMnd)
152	117	ad2antrr 736	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑖 ∈ 𝐼) → 𝑏 ∈ Fin)
153		nn0subm 21481	. . . . . . . . . . . . . . . 16 ⊢ ℕ0 ∈ (SubMnd‘ℂfld)
154	153	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑖 ∈ 𝐼) → ℕ0 ∈ (SubMnd‘ℂfld))
155	26	ad4antr 742	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑥 ∈ 𝑏) → 𝐼 ∈ 𝑉)
156	89	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑥 ∈ 𝑏) → ℕ0 ∈ V)
157	5	ssrab3 4036	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 ⊆ (ℕ0 ↑m 𝐼)
158	1	ad2antrr 736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → 𝐹:𝐴⟶𝐷)
159	158	ad2antrr 736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑥 ∈ 𝑏) → 𝐹:𝐴⟶𝐷)
160		simpllr 785	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑖 ∈ 𝐼) → 𝑏 ⊆ 𝐴)
161	160	sselda 3937	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑥 ∈ 𝑏) → 𝑥 ∈ 𝐴)
162	159, 161	ffvelcdmd 7066	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑥 ∈ 𝑏) → (𝐹‘𝑥) ∈ 𝐷)
163	157, 162	sselid 3935	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑥 ∈ 𝑏) → (𝐹‘𝑥) ∈ (ℕ0 ↑m 𝐼))
164	155, 156, 163	elmaprd 32888	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑥 ∈ 𝑏) → (𝐹‘𝑥):𝐼⟶ℕ0)
165		simplr 778	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑥 ∈ 𝑏) → 𝑖 ∈ 𝐼)
166	164, 165	ffvelcdmd 7066	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑖 ∈ 𝐼) ∧ 𝑥 ∈ 𝑏) → ((𝐹‘𝑥)‘𝑖) ∈ ℕ0)
167	166	fmpttd 7096	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑖 ∈ 𝐼) → (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)):𝑏⟶ℕ0)
168	91	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑖 ∈ 𝐼) → 0 ∈ ℕ0)
169	167, 152, 168	fdmfifsupp 9319	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑖 ∈ 𝐼) → (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)) finSupp 0)
170	73, 151, 152, 154, 167, 169	gsumsubmcl 19969	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑖 ∈ 𝐼) → (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖))) ∈ ℕ0)
171	170	fmpttd 7096	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))):𝐼⟶ℕ0)
172	143, 140, 171	elmapdd 8822	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))) ∈ (ℕ0 ↑m 𝐼))
173	91	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → 0 ∈ ℕ0)
174	171	ffund 6696	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → Fun (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))
175	117	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → 𝑏 ∈ Fin)
176	140	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑥 ∈ 𝑏) → 𝐼 ∈ 𝑉)
177	89	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑥 ∈ 𝑏) → ℕ0 ∈ V)
178	158	adantr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑥 ∈ 𝑏) → 𝐹:𝐴⟶𝐷)
179		simplr 778	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → 𝑏 ⊆ 𝐴)
180	179	sselda 3937	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑥 ∈ 𝑏) → 𝑥 ∈ 𝐴)
181	178, 180	ffvelcdmd 7066	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑥 ∈ 𝑏) → (𝐹‘𝑥) ∈ 𝐷)
182	157, 181	sselid 3935	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑥 ∈ 𝑏) → (𝐹‘𝑥) ∈ (ℕ0 ↑m 𝐼))
183	176, 177, 182	elmaprd 32888	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑥 ∈ 𝑏) → (𝐹‘𝑥):𝐼⟶ℕ0)
184	183	feqmptd 6935	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑥 ∈ 𝑏) → (𝐹‘𝑥) = (𝑖 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑖)))
185	184	oveq1d 7411	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑥 ∈ 𝑏) → ((𝐹‘𝑥) supp 0) = ((𝑖 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑖)) supp 0))
186		breq1 5104	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℎ = (𝐹‘𝑥) → (ℎ finSupp 0 ↔ (𝐹‘𝑥) finSupp 0))
187	181, 5	eleqtrdi 2873	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑥 ∈ 𝑏) → (𝐹‘𝑥) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
188	186, 187	elrabrd 3654	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑥 ∈ 𝑏) → (𝐹‘𝑥) finSupp 0)
189	188	fsuppimpd 9313	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑥 ∈ 𝑏) → ((𝐹‘𝑥) supp 0) ∈ Fin)
190	185, 189	eqeltrrd 2864	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑥 ∈ 𝑏) → ((𝑖 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑖)) supp 0) ∈ Fin)
191	190	ralrimiva 3155	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → ∀𝑥 ∈ 𝑏 ((𝑖 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑖)) supp 0) ∈ Fin)
192		iunfi 9284	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ Fin ∧ ∀𝑥 ∈ 𝑏 ((𝑖 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑖)) supp 0) ∈ Fin) → ∪ 𝑥 ∈ 𝑏 ((𝑖 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑖)) supp 0) ∈ Fin)
193	175, 191, 192	syl2anc 593	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → ∪ 𝑥 ∈ 𝑏 ((𝑖 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑖)) supp 0) ∈ Fin)
194		cmnmnd 19847	. . . . . . . . . . . . . . . . 17 ⊢ (ℂfld ∈ CMnd → ℂfld ∈ Mnd)
195	150, 194	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ℂfld ∈ Mnd
196	195	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → ℂfld ∈ Mnd)
197	115	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → 𝐴 ∈ Fin)
198	197, 179	ssexd 5281	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → 𝑏 ∈ V)
199	73, 196, 198, 140, 166	suppgsumssiun 33258	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → ((𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))) supp 0) ⊆ ∪ 𝑥 ∈ 𝑏 ((𝑖 ∈ 𝐼 ↦ ((𝐹‘𝑥)‘𝑖)) supp 0))
200	193, 199	ssfid 9213	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → ((𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))) supp 0) ∈ Fin)
201	172, 173, 174, 200	isfsuppd 9310	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))) finSupp 0)
202	142, 172, 201	elrabd 3653	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
203	202, 5	eleqtrrdi 2874	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))) ∈ 𝐷)
204		difssd 4091	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → (𝐴 ∖ 𝑏) ⊆ 𝐴)
205	204	sselda 3937	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → 𝑓 ∈ 𝐴)
206	158, 205	ffvelcdmd 7066	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → (𝐹‘𝑓) ∈ 𝐷)
207	23, 25, 16, 10, 5, 140, 141, 203, 108, 206, 30	psrmonmul2 33850	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → ((𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))(.r‘𝑆)(𝐺‘(𝐹‘𝑓))) = (𝐺‘((𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))) ∘f + (𝐹‘𝑓))))
208	171	ffnd 6692	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))) Fn 𝐼)
209	157, 206	sselid 3935	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → (𝐹‘𝑓) ∈ (ℕ0 ↑m 𝐼))
210	140, 143, 209	elmaprd 32888	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → (𝐹‘𝑓):𝐼⟶ℕ0)
211	210	ffnd 6692	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → (𝐹‘𝑓) Fn 𝐼)
212		nfv 1935	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏))
213		ovexd 7431	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑖 ∈ 𝐼) → (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖))) ∈ V)
214		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖)))) = (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖))))
215	212, 213, 214	fnmptd 6662	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖)))) Fn 𝐼)
216		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))) = (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖))))
217		fveq2 6867	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → ((𝐹‘𝑥)‘𝑖) = ((𝐹‘𝑥)‘𝑗))
218	217	mpteq2dv 5195	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)) = (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑗)))
219	218	oveq2d 7412	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖))) = (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑗))))
220		simpr 488	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) → 𝑗 ∈ 𝐼)
221		ovexd 7431	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) → (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑗))) ∈ V)
222	216, 219, 220, 221	fvmptd3 6999	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) → ((𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖))))‘𝑗) = (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑗))))
223		eqidd 2764	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) → ((𝐹‘𝑓)‘𝑗) = ((𝐹‘𝑓)‘𝑗))
224	217	mpteq2dv 5195	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖)) = (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑗)))
225	224	oveq2d 7412	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖))) = (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑗))))
226		ovexd 7431	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) → (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑗))) ∈ V)
227	214, 225, 220, 226	fvmptd3 6999	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) → ((𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖))))‘𝑗) = (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑗))))
228		cnfldbas 21435	. . . . . . . . . . . . 13 ⊢ ℂ = (Base‘ℂfld)
229		cnfldadd 21437	. . . . . . . . . . . . 13 ⊢ + = (+g‘ℂfld)
230	150	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) → ℂfld ∈ CMnd)
231	175	adantr 484	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) → 𝑏 ∈ Fin)
232	183	adantlr 725	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) ∧ 𝑥 ∈ 𝑏) → (𝐹‘𝑥):𝐼⟶ℕ0)
233		nn0sscn 12496	. . . . . . . . . . . . . . . 16 ⊢ ℕ0 ⊆ ℂ
234	233	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) ∧ 𝑥 ∈ 𝑏) → ℕ0 ⊆ ℂ)
235	232, 234	fssd 6709	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) ∧ 𝑥 ∈ 𝑏) → (𝐹‘𝑥):𝐼⟶ℂ)
236		simplr 778	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) ∧ 𝑥 ∈ 𝑏) → 𝑗 ∈ 𝐼)
237	235, 236	ffvelcdmd 7066	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) ∧ 𝑥 ∈ 𝑏) → ((𝐹‘𝑥)‘𝑗) ∈ ℂ)
238		simplr 778	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) → 𝑓 ∈ (𝐴 ∖ 𝑏))
239	238	eldifbd 3918	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) → ¬ 𝑓 ∈ 𝑏)
240	210	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) → (𝐹‘𝑓):𝐼⟶ℕ0)
241	233	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) → ℕ0 ⊆ ℂ)
242	240, 241	fssd 6709	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) → (𝐹‘𝑓):𝐼⟶ℂ)
243	242, 220	ffvelcdmd 7066	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) → ((𝐹‘𝑓)‘𝑗) ∈ ℂ)
244		fveq2 6867	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑓 → (𝐹‘𝑥) = (𝐹‘𝑓))
245	244	fveq1d 6869	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑓 → ((𝐹‘𝑥)‘𝑗) = ((𝐹‘𝑓)‘𝑗))
246	228, 229, 230, 231, 237, 238, 239, 243, 245	gsumunsn 20010	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) → (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑗))) = ((ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑗))) + ((𝐹‘𝑓)‘𝑗)))
247	227, 246	eqtr2d 2799	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ 𝑗 ∈ 𝐼) → ((ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑗))) + ((𝐹‘𝑓)‘𝑗)) = ((𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖))))‘𝑗))
248	140, 208, 211, 215, 222, 223, 247	offveq 7686	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → ((𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))) ∘f + (𝐹‘𝑓)) = (𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖)))))
249	248	fveq2d 6871	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → (𝐺‘((𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))) ∘f + (𝐹‘𝑓))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖))))))
250	207, 249	eqtrd 2798	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → ((𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))(.r‘𝑆)(𝐺‘(𝐹‘𝑓))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖))))))
251	250	adantr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))) → ((𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))(.r‘𝑆)(𝐺‘(𝐹‘𝑓))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖))))))
252	133, 139, 251	3eqtrd 2802	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))) → (𝑀 Σg (𝑙 ∈ (𝑏 ∪ {𝑓}) ↦ (𝐺‘(𝐹‘𝑙)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖))))))
253	106, 252	eqtrid 2810	. . . . 5 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) ∧ (𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖)))))) → (𝑀 Σg (𝑘 ∈ (𝑏 ∪ {𝑓}) ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖))))))
254	253	ex 416	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑓 ∈ (𝐴 ∖ 𝑏)) → ((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖))))) → (𝑀 Σg (𝑘 ∈ (𝑏 ∪ {𝑓}) ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖)))))))
255	254	anasss 470	. . 3 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑓 ∈ (𝐴 ∖ 𝑏))) → ((𝑀 Σg (𝑘 ∈ 𝑏 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝑏 ↦ ((𝐹‘𝑥)‘𝑖))))) → (𝑀 Σg (𝑘 ∈ (𝑏 ∪ {𝑓}) ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ (𝑏 ∪ {𝑓}) ↦ ((𝐹‘𝑥)‘𝑖)))))))
256	42, 49, 56, 63, 103, 255, 114	findcard2d 9135	. 2 ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝐴 ↦ (𝐺‘(𝐹‘𝑘)))) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)‘𝑖))))))
257	35, 256	eqtrd 2798	1 ⊢ (𝜑 → (𝑀 Σg (𝐺 ∘ 𝐹)) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)‘𝑖))))))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1561   ∈ wcel 2143  ∀wral 3077  {crab 3415  Vcvv 3455   ∖ cdif 3902   ∪ cun 3903   ⊆ wss 3905  ∅c0 4286  ifcif 4481  {csn 4583  ∪ ciun 4950   class class class wbr 5101   ↦ cmpt 5182   × cxp 5646   ∘ ccom 5652  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396   ∘_f cof 7658   supp csupp 8140   ↑_m cmap 8808  Fincfn 8927   finSupp cfsupp 9305  ℂcc 11082  0cc0 11084   + caddc 11087  ℕ₀cn0 12491  Basecbs 17255  ._rcmulr 17297  0_gc0g 17478   Σ_g cgsu 17479  Mndcmnd 18778  SubMndcsubmnd 18826  Grpcgrp 18985  CMndccmn 19830  mulGrpcmgp 20196  1_rcur 20241  Ringcrg 20293  CRingccrg 20294  Fieldcfield 20789  ℂ_fldccnfld 21431   mPwSer cmps 21963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718  ax-cnex 11140  ax-resscn 11141  ax-1cn 11142  ax-icn 11143  ax-addcl 11144  ax-addrcl 11145  ax-mulcl 11146  ax-mulrcl 11147  ax-mulcom 11148  ax-addass 11149  ax-mulass 11150  ax-distr 11151  ax-i2m1 11152  ax-1ne0 11153  ax-1rid 11154  ax-rnegex 11155  ax-rrecex 11156  ax-cnre 11157  ax-pre-lttri 11158  ax-pre-lttrn 11159  ax-pre-ltadd 11160  ax-pre-mulgt0 11161  ax-addf 11163

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-se 5602  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7660  df-ofr 7661  df-om 7847  df-1st 7970  df-2nd 7971  df-supp 8141  df-tpos 8206  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8678  df-map 8810  df-pm 8811  df-ixp 8880  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9306  df-sup 9386  df-oi 9456  df-card 9909  df-pnf 11229  df-mnf 11230  df-xr 11231  df-ltxr 11232  df-le 11233  df-sub 11427  df-neg 11428  df-div 11856  df-nn 12221  df-2 12290  df-3 12291  df-4 12292  df-5 12293  df-6 12294  df-7 12295  df-8 12296  df-9 12297  df-n0 12492  df-z 12579  df-dec 12699  df-uz 12850  df-fz 13523  df-fzo 13670  df-seq 14025  df-hash 14354  df-struct 17193  df-sets 17210  df-slot 17228  df-ndx 17240  df-base 17256  df-ress 17277  df-plusg 17309  df-mulr 17310  df-starv 17311  df-sca 17312  df-vsca 17313  df-ip 17314  df-tset 17315  df-ple 17316  df-ds 17318  df-unif 17319  df-hom 17320  df-cco 17321  df-0g 17480  df-gsum 17481  df-prds 17486  df-pws 17488  df-mre 17624  df-mrc 17625  df-acs 17627  df-mgm 18684  df-sgrp 18763  df-mnd 18779  df-mhm 18827  df-submnd 18828  df-grp 18988  df-minusg 18989  df-mulg 19120  df-ghm 19264  df-cntz 19367  df-cmn 19832  df-abl 19833  df-mgp 20197  df-rng 20209  df-ur 20242  df-ring 20295  df-cring 20296  df-oppr 20396  df-dvdsr 20416  df-unit 20417  df-invr 20447  df-dvr 20460  df-drng 20790  df-field 20791  df-cnfld 21432  df-psr 21968

This theorem is referenced by:  mplmonprod  33853