Theorem psropprmul 19608

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 2622	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2622	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
3		ringcmn 18581	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
4	3	3ad2ant1 1082	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝑅 ∈ CMnd)
5	4	adantr 481	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → 𝑅 ∈ CMnd)
6		ovex 6678	. . . . . . . 8 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V
7	6	rabex 4813	. . . . . . 7 ⊢ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∈ V
8	7	rabex 4813	. . . . . 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 1100	. . . . . . 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 2622	. . . . . . . . . 10 ⊢ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} = {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}
13		psropprmul.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑌)
14		simp3 1063	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵)
15	11, 1, 12, 13, 14	psrelbas 19379	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺:{𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
16	15	adantr 481	. . . . . . . 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 3359	. . . . . . . 8 ⊢ (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} → 𝑒 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin})
18		ffvelrn 6357	. . . . . . . 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 494	. . . . . . 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 1062	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵)
21	11, 1, 12, 13, 20	psrelbas 19379	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹:{𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
22	21	ad2antrr 762	. . . . . . . 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 3687	. . . . . . . . 9 ⊢ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} ⊆ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}
24		reldmpsr 19361	. . . . . . . . . . . . 13 ⊢ Rel dom mPwSer
25	11, 13, 24	strov2rcl 15922	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ 𝐵 → 𝐼 ∈ V)
26	25	3ad2ant3 1084	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐼 ∈ V)
27	26	ad2antrr 762	. . . . . . . . . 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 792	. . . . . . . . . 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 477	. . . . . . . . . 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 2622	. . . . . . . . . . 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 19373	. . . . . . . . . 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 1326	. . . . . . . . 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 3601	. . . . . . . 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 6360	. . . . . . 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 2622	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
36	1, 35	ringcl 18561	. . . . . . 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 1326	. . . . . 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 2622	. . . . . 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 6385	. . . . 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 6484	. . . . . . 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 5926	. . . . . . 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 6201	. . . . . . 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 19372	. . . . . . 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 488	. . . . . 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 7308	. . . . . . . 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 5631	. . . . . . . 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 3612	. . . . . . 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 8290	. . . . . 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 1334	. . . . 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 19374	. . . . . 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 488	. . . . 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 18317	. . . 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 762	. . . . . . . 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 792	. . . . . . . 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 477	. . . . . . . 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 19373	. . . . . . . 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 1326	. . . . . . 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 2623	. . . . . . 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 2623	. . . . . . 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 6191	. . . . . . . 8 ⊢ (𝑒 = (𝑏 ∘𝑓 − 𝑐) → (𝐺‘𝑒) = (𝐺‘(𝑏 ∘𝑓 − 𝑐)))
65		oveq2 6658	. . . . . . . . 9 ⊢ (𝑒 = (𝑏 ∘𝑓 − 𝑐) → (𝑏 ∘𝑓 − 𝑒) = (𝑏 ∘𝑓 − (𝑏 ∘𝑓 − 𝑐)))
66	65	fveq2d 6195	. . . . . . . 8 ⊢ (𝑒 = (𝑏 ∘𝑓 − 𝑐) → (𝐹‘(𝑏 ∘𝑓 − 𝑒)) = (𝐹‘(𝑏 ∘𝑓 − (𝑏 ∘𝑓 − 𝑐))))
67	64, 66	oveq12d 6668	. . . . . . 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 6396	. . . . . 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 19365	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ V ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → 𝑏:𝐼⟶ℕ0)
70	26, 69	sylan 488	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → 𝑏:𝐼⟶ℕ0)
71	70	adantr 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}) → 𝑏:𝐼⟶ℕ0)
72	26	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → 𝐼 ∈ V)
73		elrabi 3359	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘𝑟 ≤ 𝑏} → 𝑐 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin})
74	12	psrbagf 19365	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ V ∧ 𝑐 ∈ {𝑎 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → 𝑐:𝐼⟶ℕ0)
75	72, 73, 74	syl2an 494	. . . . . . . . . . 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 11302	. . . . . . . . . . . . 13 ⊢ (𝑒 ∈ ℕ0 → 𝑒 ∈ ℂ)
77		nn0cn 11302	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ℕ0 → 𝑓 ∈ ℂ)
78		nncan 10310	. . . . . . . . . . . . 13 ⊢ ((𝑒 ∈ ℂ ∧ 𝑓 ∈ ℂ) → (𝑒 − (𝑒 − 𝑓)) = 𝑓)
79	76, 77, 78	syl2an 494	. . . . . . . . . . . 12 ⊢ ((𝑒 ∈ ℕ0 ∧ 𝑓 ∈ ℕ0) → (𝑒 − (𝑒 − 𝑓)) = 𝑓)
80	79	adantl 482	. . . . . . . . . . 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 6935	. . . . . . . . . 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 6195	. . . . . . . . 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 6666	. . . . . . . 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 2622	. . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆)
86	1, 35, 84, 85	opprmul 18626	. . . . . . . 8 ⊢ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘𝑓 − 𝑐))) = ((𝐺‘(𝑏 ∘𝑓 − 𝑐))(.r‘𝑅)(𝐹‘𝑐))
87	83, 86	syl6eqr 2674	. . . . . . 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 4744	. . . . . 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 2656	. . . . 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 6666	. . . 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 6486	. . . . . . . 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 6201	. . . . . . . . 9 ⊢ (oppr‘𝑅) ∈ V
95	84, 94	eqeltri 2697	. . . . . . . 8 ⊢ 𝑆 ∈ V
96	95	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑆 ∈ V)
97	84, 1	opprbas 18629	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑆)
98	97	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑆))
99		eqid 2622	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅)
100	84, 99	oppradd 18630	. . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑆)
101	100	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (+g‘𝑅) = (+g‘𝑆))
102	92, 93, 96, 98, 101	gsumpropd 17272	. . . . . 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 1082	. . . . 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 481	. . . 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 2660	. . 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 4744	. 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 19385	. 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 2622	. . 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 19605	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆)))
114	11	fveq2i 6194	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘(𝐼 mPwSer 𝑅))
115	13, 114	eqtri 2644	. . . . 5 ⊢ 𝐵 = (Base‘(𝐼 mPwSer 𝑅))
116	109	fveq2i 6194	. . . . 5 ⊢ (Base‘𝑍) = (Base‘(𝐼 mPwSer 𝑆))
117	113, 115, 116	3eqtr4g 2681	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐵 = (Base‘𝑍))
118	20, 117	eleqtrd 2703	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ (Base‘𝑍))
119	14, 117	eleqtrd 2703	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ (Base‘𝑍))
120	109, 110, 85, 111, 12, 118, 119	psrmulfval 19385	. 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 2667	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) = (𝐺 · 𝐹))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200   ⊆ wss 3574   class class class wbr 4653   ↦ cmpt 4729  ^◡ccnv 5113  dom cdm 5114   “ cima 5117   ∘ ccom 5118  Fun wfun 5882  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895   ∘_𝑟 cofr 6896   supp csupp 7295   ↑_𝑚 cmap 7857  Fincfn 7955   finSupp cfsupp 8275  ℂcc 9934   ≤ cle 10075   − cmin 10266  ℕcn 11020  ℕ₀cn0 11292  Basecbs 15857  +_gcplusg 15941  ._rcmulr 15942  0_gc0g 16100   Σ_g cgsu 16101  CMndccmn 18193  Ringcrg 18547  opp_rcoppr 18622   mPwSer cmps 19351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-tset 15960  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-psr 19356

This theorem is referenced by:  ply1opprmul  19609