Theorem psropprmul 19429

Description: Reversing multiplication in a ring reverses multiplication in the power series ring. (Contributed by Stefan O'Rear, 27-Mar-2015.)

Hypotheses

Ref	Expression
psropprmul.y	⊢ 𝑌 = (𝐼 mPwSer 𝑅)
psropprmul.s	⊢ 𝑆 = (oppr‘𝑅)
psropprmul.z	⊢ 𝑍 = (𝐼 mPwSer 𝑆)
psropprmul.t	⊢ · = (.r‘𝑌)
psropprmul.u	⊢ ∙ = (.r‘𝑍)
psropprmul.b	⊢ 𝐵 = (Base‘𝑌)

Assertion

Ref	Expression
psropprmul	⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) = (𝐺 · 𝐹))

Proof of Theorem psropprmul

Dummy variables 𝑏 𝑐 𝑒 𝑓 𝑎 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2610	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2610	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
3		ringcmn 18404	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
4	3	3ad2ant1 1075	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝑅 ∈ CMnd)
5	4	adantr 480	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → 𝑅 ∈ CMnd)
6		ovex 6577	. . . . . . . 8 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V
7	6	rabex 4740	. . . . . . 7 ⊢ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∈ V
8	7	rabex 4740	. . . . . 6 ⊢ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ∈ V
9	8	a1i 11	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ∈ V)
10		simpll1 1093	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → 𝑅 ∈ Ring)
11		psropprmul.y	. . . . . . . . . 10 ⊢ 𝑌 = (𝐼 mPwSer 𝑅)
12		eqid 2610	. . . . . . . . . 10 ⊢ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} = {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}
13		psropprmul.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑌)
14		simp3 1056	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵)
15	11, 1, 12, 13, 14	psrelbas 19200	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺:{𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
16	15	adantr 480	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → 𝐺:{𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
17		elrabi 3328	. . . . . . . 8 ⊢ (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} → 𝑒 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin})
18		ffvelrn 6265	. . . . . . . 8 ⊢ ((𝐺:{𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}⟶(Base‘𝑅) ∧ 𝑒 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → (𝐺‘𝑒) ∈ (Base‘𝑅))
19	16, 17, 18	syl2an 493	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → (𝐺‘𝑒) ∈ (Base‘𝑅))
20		simp2 1055	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵)
21	11, 1, 12, 13, 20	psrelbas 19200	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹:{𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
22	21	ad2antrr 758	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → 𝐹:{𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
23		ssrab2 3650	. . . . . . . . 9 ⊢ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ⊆ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}
24		reldmpsr 19182	. . . . . . . . . . . . 13 ⊢ Rel dom mPwSer
25	11, 13, 24	strov2rcl 15750	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ 𝐵 → 𝐼 ∈ V)
26	25	3ad2ant3 1077	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐼 ∈ V)
27	26	ad2antrr 758	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → 𝐼 ∈ V)
28		simplr 788	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin})
29		simpr 476	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → 𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏})
30		eqid 2610	. . . . . . . . . . 11 ⊢ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} = {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}
31	12, 30	psrbagconcl 19194	. . . . . . . . . 10 ⊢ ((𝐼 ∈ V ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∧ 𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → (𝑏 ∘𝑓 − 𝑒) ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏})
32	27, 28, 29, 31	syl3anc 1318	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → (𝑏 ∘𝑓 − 𝑒) ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏})
33	23, 32	sseldi 3566	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → (𝑏 ∘𝑓 − 𝑒) ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin})
34	22, 33	ffvelrnd 6268	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → (𝐹‘(𝑏 ∘𝑓 − 𝑒)) ∈ (Base‘𝑅))
35		eqid 2610	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
36	1, 35	ringcl 18384	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘𝑒) ∈ (Base‘𝑅) ∧ (𝐹‘(𝑏 ∘𝑓 − 𝑒)) ∈ (Base‘𝑅)) → ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒))) ∈ (Base‘𝑅))
37	10, 19, 34, 36	syl3anc 1318	. . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒))) ∈ (Base‘𝑅))
38		eqid 2610	. . . . . 6 ⊢ (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))) = (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒))))
39	37, 38	fmptd 6292	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))):{𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}⟶(Base‘𝑅))
40		mptexg 6389	. . . . . . 7 ⊢ ({𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ∈ V → (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))) ∈ V)
41	8, 40	mp1i 13	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))) ∈ V)
42		funmpt 5840	. . . . . . 7 ⊢ Fun (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒))))
43	42	a1i 11	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → Fun (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))))
44		fvex 6113	. . . . . . 7 ⊢ (0g‘𝑅) ∈ V
45	44	a1i 11	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → (0g‘𝑅) ∈ V)
46	12	psrbaglefi 19193	. . . . . . 7 ⊢ ((𝐼 ∈ V ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ∈ Fin)
47	26, 46	sylan 487	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ∈ Fin)
48		suppssdm 7195	. . . . . . . 8 ⊢ ((𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))) supp (0g‘𝑅)) ⊆ dom (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒))))
49	38	dmmptss 5548	. . . . . . . 8 ⊢ dom (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))) ⊆ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}
50	48, 49	sstri 3577	. . . . . . 7 ⊢ ((𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))) supp (0g‘𝑅)) ⊆ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}
51	50	a1i 11	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → ((𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))) supp (0g‘𝑅)) ⊆ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏})
52		suppssfifsupp 8173	. . . . . 6 ⊢ ((((𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))) ∈ V ∧ Fun (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))) ∧ (0g‘𝑅) ∈ V) ∧ ({𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ∈ Fin ∧ ((𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))) supp (0g‘𝑅)) ⊆ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏})) → (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))) finSupp (0g‘𝑅))
53	41, 43, 45, 47, 51, 52	syl32anc 1326	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))) finSupp (0g‘𝑅))
54	12, 30	psrbagconf1o 19195	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ (𝑏 ∘𝑓 − 𝑐)):{𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}–1-1-onto→{𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏})
55	26, 54	sylan 487	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ (𝑏 ∘𝑓 − 𝑐)):{𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}–1-1-onto→{𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏})
56	1, 2, 5, 9, 39, 53, 55	gsumf1o 18140	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒))))) = (𝑅 Σg ((𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))) ∘ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ (𝑏 ∘𝑓 − 𝑐)))))
57	26	ad2antrr 758	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → 𝐼 ∈ V)
58		simplr 788	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin})
59		simpr 476	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏})
60	12, 30	psrbagconcl 19194	. . . . . . . 8 ⊢ ((𝐼 ∈ V ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → (𝑏 ∘𝑓 − 𝑐) ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏})
61	57, 58, 59, 60	syl3anc 1318	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → (𝑏 ∘𝑓 − 𝑐) ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏})
62		eqidd 2611	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ (𝑏 ∘𝑓 − 𝑐)) = (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ (𝑏 ∘𝑓 − 𝑐)))
63		eqidd 2611	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))) = (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))))
64		fveq2 6103	. . . . . . . 8 ⊢ (𝑒 = (𝑏 ∘𝑓 − 𝑐) → (𝐺‘𝑒) = (𝐺‘(𝑏 ∘𝑓 − 𝑐)))
65		oveq2 6557	. . . . . . . . 9 ⊢ (𝑒 = (𝑏 ∘𝑓 − 𝑐) → (𝑏 ∘𝑓 − 𝑒) = (𝑏 ∘𝑓 − (𝑏 ∘𝑓 − 𝑐)))
66	65	fveq2d 6107	. . . . . . . 8 ⊢ (𝑒 = (𝑏 ∘𝑓 − 𝑐) → (𝐹‘(𝑏 ∘𝑓 − 𝑒)) = (𝐹‘(𝑏 ∘𝑓 − (𝑏 ∘𝑓 − 𝑐))))
67	64, 66	oveq12d 6567	. . . . . . 7 ⊢ (𝑒 = (𝑏 ∘𝑓 − 𝑐) → ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒))) = ((𝐺‘(𝑏 ∘𝑓 − 𝑐))(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − (𝑏 ∘𝑓 − 𝑐)))))
68	61, 62, 63, 67	fmptco 6303	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → ((𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))) ∘ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ (𝑏 ∘𝑓 − 𝑐))) = (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘(𝑏 ∘𝑓 − 𝑐))(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − (𝑏 ∘𝑓 − 𝑐))))))
69	12	psrbagf 19186	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ V ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → 𝑏:𝐼⟶ℕ0)
70	26, 69	sylan 487	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → 𝑏:𝐼⟶ℕ0)
71	70	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → 𝑏:𝐼⟶ℕ0)
72	26	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → 𝐼 ∈ V)
73		elrabi 3328	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} → 𝑐 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin})
74	12	psrbagf 19186	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ V ∧ 𝑐 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → 𝑐:𝐼⟶ℕ0)
75	72, 73, 74	syl2an 493	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → 𝑐:𝐼⟶ℕ0)
76		nn0cn 11179	. . . . . . . . . . . . 13 ⊢ (𝑒 ∈ ℕ0 → 𝑒 ∈ ℂ)
77		nn0cn 11179	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ℕ0 → 𝑓 ∈ ℂ)
78		nncan 10189	. . . . . . . . . . . . 13 ⊢ ((𝑒 ∈ ℂ ∧ 𝑓 ∈ ℂ) → (𝑒 − (𝑒 − 𝑓)) = 𝑓)
79	76, 77, 78	syl2an 493	. . . . . . . . . . . 12 ⊢ ((𝑒 ∈ ℕ0 ∧ 𝑓 ∈ ℕ0) → (𝑒 − (𝑒 − 𝑓)) = 𝑓)
80	79	adantl 481	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) ∧ (𝑒 ∈ ℕ0 ∧ 𝑓 ∈ ℕ0)) → (𝑒 − (𝑒 − 𝑓)) = 𝑓)
81	57, 71, 75, 80	caonncan 6833	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → (𝑏 ∘𝑓 − (𝑏 ∘𝑓 − 𝑐)) = 𝑐)
82	81	fveq2d 6107	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → (𝐹‘(𝑏 ∘𝑓 − (𝑏 ∘𝑓 − 𝑐))) = (𝐹‘𝑐))
83	82	oveq2d 6565	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → ((𝐺‘(𝑏 ∘𝑓 − 𝑐))(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − (𝑏 ∘𝑓 − 𝑐)))) = ((𝐺‘(𝑏 ∘𝑓 − 𝑐))(.r‘𝑅)(𝐹‘𝑐)))
84		psropprmul.s	. . . . . . . . 9 ⊢ 𝑆 = (oppr‘𝑅)
85		eqid 2610	. . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆)
86	1, 35, 84, 85	opprmul 18449	. . . . . . . 8 ⊢ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘𝑓 − 𝑐))) = ((𝐺‘(𝑏 ∘𝑓 − 𝑐))(.r‘𝑅)(𝐹‘𝑐))
87	83, 86	syl6eqr 2662	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏}) → ((𝐺‘(𝑏 ∘𝑓 − 𝑐))(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − (𝑏 ∘𝑓 − 𝑐)))) = ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘𝑓 − 𝑐))))
88	87	mpteq2dva 4672	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘(𝑏 ∘𝑓 − 𝑐))(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − (𝑏 ∘𝑓 − 𝑐))))) = (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘𝑓 − 𝑐)))))
89	68, 88	eqtrd 2644	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → ((𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))) ∘ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ (𝑏 ∘𝑓 − 𝑐))) = (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘𝑓 − 𝑐)))))
90	89	oveq2d 6565	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → (𝑅 Σg ((𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))) ∘ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ (𝑏 ∘𝑓 − 𝑐)))) = (𝑅 Σg (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘𝑓 − 𝑐))))))
91	8	mptex 6390	. . . . . . . 8 ⊢ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘𝑓 − 𝑐)))) ∈ V
92	91	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘𝑓 − 𝑐)))) ∈ V)
93		id 22	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Ring)
94		fvex 6113	. . . . . . . . 9 ⊢ (oppr‘𝑅) ∈ V
95	84, 94	eqeltri 2684	. . . . . . . 8 ⊢ 𝑆 ∈ V
96	95	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑆 ∈ V)
97	84, 1	opprbas 18452	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑆)
98	97	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑆))
99		eqid 2610	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅)
100	84, 99	oppradd 18453	. . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑆)
101	100	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (+g‘𝑅) = (+g‘𝑆))
102	92, 93, 96, 98, 101	gsumpropd 17095	. . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑅 Σg (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘𝑓 − 𝑐))))) = (𝑆 Σg (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘𝑓 − 𝑐))))))
103	102	3ad2ant1 1075	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝑅 Σg (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘𝑓 − 𝑐))))) = (𝑆 Σg (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘𝑓 − 𝑐))))))
104	103	adantr 480	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘𝑓 − 𝑐))))) = (𝑆 Σg (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘𝑓 − 𝑐))))))
105	56, 90, 104	3eqtrd 2648	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒))))) = (𝑆 Σg (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘𝑓 − 𝑐))))))
106	105	mpteq2dva 4672	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))))) = (𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (𝑆 Σg (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘𝑓 − 𝑐)))))))
107		psropprmul.t	. . 3 ⊢ · = (.r‘𝑌)
108	11, 13, 35, 107, 12, 14, 20	psrmulfval 19206	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐺 · 𝐹) = (𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘𝑓 − 𝑒)))))))
109		psropprmul.z	. . 3 ⊢ 𝑍 = (𝐼 mPwSer 𝑆)
110		eqid 2610	. . 3 ⊢ (Base‘𝑍) = (Base‘𝑍)
111		psropprmul.u	. . 3 ⊢ ∙ = (.r‘𝑍)
112	97	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (Base‘𝑅) = (Base‘𝑆))
113	112	psrbaspropd 19426	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆)))
114	11	fveq2i 6106	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘(𝐼 mPwSer 𝑅))
115	13, 114	eqtri 2632	. . . . 5 ⊢ 𝐵 = (Base‘(𝐼 mPwSer 𝑅))
116	109	fveq2i 6106	. . . . 5 ⊢ (Base‘𝑍) = (Base‘(𝐼 mPwSer 𝑆))
117	113, 115, 116	3eqtr4g 2669	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐵 = (Base‘𝑍))
118	20, 117	eleqtrd 2690	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ (Base‘𝑍))
119	14, 117	eleqtrd 2690	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ (Base‘𝑍))
120	109, 110, 85, 111, 12, 118, 119	psrmulfval 19206	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) = (𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (𝑆 Σg (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ↦ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘𝑓 − 𝑐)))))))
121	106, 108, 120	3eqtr4rd 2655	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) = (𝐺 · 𝐹))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643  ^◡ccnv 5037  dom cdm 5038   “ cima 5041   ∘ ccom 5042  Fun wfun 5798  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793   ∘_𝑟 cofr 6794   supp csupp 7182   ↑_𝑚 cmap 7744  Fincfn 7841   finSupp cfsupp 8158  ℂcc 9813   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℕ₀cn0 11169  Basecbs 15695  +_gcplusg 15768  ._rcmulr 15769  0_gc0g 15923   Σ_g cgsu 15924  CMndccmn 18016  Ringcrg 18370  opp_rcoppr 18445   mPwSer cmps 19172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-tset 15787  df-0g 15925  df-gsum 15926  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-oppr 18446  df-psr 19177

This theorem is referenced by:  ply1opprmul  19430