Theorem psropprmul 22122

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 2729	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2729	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
3		ringcmn 20191	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
4	3	3ad2ant1 1133	. . . . . 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 7420	. . . . . . . 8 ⊢ (ℕ0 ↑m 𝐼) ∈ V
7	6	rabex 5294	. . . . . . 7 ⊢ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∈ V
8	7	rabex 5294	. . . . . 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 1213	. . . . . . 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 2729	. . . . . . . . . 10 ⊢ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} = {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}
13		psropprmul.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑌)
14		simp3 1138	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵)
15	11, 1, 12, 13, 14	psrelbas 21843	. . . . . . . . 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 3654	. . . . . . . 8 ⊢ (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘r ≤ 𝑏} → 𝑒 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin})
18		ffvelcdm 7053	. . . . . . . 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 596	. . . . . . 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 1137	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵)
21	11, 1, 12, 13, 20	psrelbas 21843	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹:{𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
22	21	ad2antrr 726	. . . . . . . 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 4043	. . . . . . . . 9 ⊢ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘r ≤ 𝑏} ⊆ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}
24		eqid 2729	. . . . . . . . . . 11 ⊢ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘r ≤ 𝑏} = {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘r ≤ 𝑏}
25	12, 24	psrbagconcl 21836	. . . . . . . . . 10 ⊢ ((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})
26	25	adantll 714	. . . . . . . . 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})
27	23, 26	sselid 3944	. . . . . . . 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})
28	22, 27	ffvelcdmd 7057	. . . . . . 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))
29		eqid 2729	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
30	1, 29	ringcl 20159	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘𝑒) ∈ (Base‘𝑅) ∧ (𝐹‘(𝑏 ∘f − 𝑒)) ∈ (Base‘𝑅)) → ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘f − 𝑒))) ∈ (Base‘𝑅))
31	10, 19, 28, 30	syl3anc 1373	. . . . . 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))
32	31	fmpttd 7087	. . . . 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))
33		mptexg 7195	. . . . . . 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)
34	8, 33	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)
35		funmpt 6554	. . . . . . 7 ⊢ Fun (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘r ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘f − 𝑒))))
36	35	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)))))
37		fvexd 6873	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → (0g‘𝑅) ∈ V)
38	12	psrbaglefi 21835	. . . . . . 7 ⊢ (𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} → {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘r ≤ 𝑏} ∈ Fin)
39	38	adantl 481	. . . . . 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)
40		suppssdm 8156	. . . . . . . 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))))
41		eqid 2729	. . . . . . . . 9 ⊢ (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))))
42	41	dmmptss 6214	. . . . . . . 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}
43	40, 42	sstri 3956	. . . . . . 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}
44	43	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})
45		suppssfifsupp 9331	. . . . . 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))
46	34, 36, 37, 39, 44, 45	syl32anc 1380	. . . . 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))
47	12, 24	psrbagconf1o 21838	. . . . . 6 ⊢ (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})
48	47	adantl 481	. . . . 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})
49	1, 2, 5, 9, 32, 46, 48	gsumf1o 19846	. . . 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)))))
50	12, 24	psrbagconcl 21836	. . . . . . . 8 ⊢ ((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})
51	50	adantll 714	. . . . . . 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})
52		eqidd 2730	. . . . . . 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)))
53		eqidd 2730	. . . . . . 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)))))
54		fveq2 6858	. . . . . . . 8 ⊢ (𝑒 = (𝑏 ∘f − 𝑐) → (𝐺‘𝑒) = (𝐺‘(𝑏 ∘f − 𝑐)))
55		oveq2 7395	. . . . . . . . 9 ⊢ (𝑒 = (𝑏 ∘f − 𝑐) → (𝑏 ∘f − 𝑒) = (𝑏 ∘f − (𝑏 ∘f − 𝑐)))
56	55	fveq2d 6862	. . . . . . . 8 ⊢ (𝑒 = (𝑏 ∘f − 𝑐) → (𝐹‘(𝑏 ∘f − 𝑒)) = (𝐹‘(𝑏 ∘f − (𝑏 ∘f − 𝑐))))
57	54, 56	oveq12d 7405	. . . . . . 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)))))
58	51, 52, 53, 57	fmptco 7101	. . . . . 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))))))
59		reldmpsr 21823	. . . . . . . . . . . . . 14 ⊢ Rel dom mPwSer
60	11, 13, 59	strov2rcl 17187	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ 𝐵 → 𝐼 ∈ V)
61	60	3ad2ant3 1135	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐼 ∈ V)
62	61	ad2antrr 726	. . . . . . . . . . 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}) → 𝐼 ∈ V)
63	12	psrbagf 21827	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} → 𝑏:𝐼⟶ℕ0)
64	63	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → 𝑏:𝐼⟶ℕ0)
65	64	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)
66		elrabi 3654	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘r ≤ 𝑏} → 𝑐 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin})
67	12	psrbagf 21827	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} → 𝑐:𝐼⟶ℕ0)
68	66, 67	syl 17	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘r ≤ 𝑏} → 𝑐:𝐼⟶ℕ0)
69	68	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}) → 𝑐:𝐼⟶ℕ0)
70		nn0cn 12452	. . . . . . . . . . . . 13 ⊢ (𝑒 ∈ ℕ0 → 𝑒 ∈ ℂ)
71		nn0cn 12452	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ℕ0 → 𝑓 ∈ ℂ)
72		nncan 11451	. . . . . . . . . . . . 13 ⊢ ((𝑒 ∈ ℂ ∧ 𝑓 ∈ ℂ) → (𝑒 − (𝑒 − 𝑓)) = 𝑓)
73	70, 71, 72	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝑒 ∈ ℕ0 ∧ 𝑓 ∈ ℕ0) → (𝑒 − (𝑒 − 𝑓)) = 𝑓)
74	73	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)
75	62, 65, 69, 74	caonncan 7697	. . . . . . . . . 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)
76	75	fveq2d 6862	. . . . . . . . 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))
77	76	oveq2d 7403	. . . . . . . 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)))
78		psropprmul.s	. . . . . . . . 9 ⊢ 𝑆 = (oppr‘𝑅)
79		eqid 2729	. . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆)
80	1, 29, 78, 79	opprmul 20249	. . . . . . . 8 ⊢ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘f − 𝑐))) = ((𝐺‘(𝑏 ∘f − 𝑐))(.r‘𝑅)(𝐹‘𝑐))
81	77, 80	eqtr4di 2782	. . . . . . 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))))
82	81	mpteq2dva 5200	. . . . . 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)))))
83	58, 82	eqtrd 2764	. . . . 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)))))
84	83	oveq2d 7403	. . . 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))))))
85	8	mptex 7197	. . . . . . . 8 ⊢ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘r ≤ 𝑏} ↦ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘f − 𝑐)))) ∈ V
86	85	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)
87		id 22	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Ring)
88	78	fvexi 6872	. . . . . . . 8 ⊢ 𝑆 ∈ V
89	88	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑆 ∈ V)
90	78, 1	opprbas 20252	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑆)
91	90	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑆))
92		eqid 2729	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅)
93	78, 92	oppradd 20253	. . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑆)
94	93	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (+g‘𝑅) = (+g‘𝑆))
95	86, 87, 89, 91, 94	gsumpropd 18605	. . . . . 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))))))
96	95	3ad2ant1 1133	. . . . 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))))))
97	96	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))))))
98	49, 84, 97	3eqtrd 2768	. . 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))))))
99	98	mpteq2dva 5200	. 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)))))))
100		psropprmul.t	. . 3 ⊢ · = (.r‘𝑌)
101	11, 13, 29, 100, 12, 14, 20	psrmulfval 21852	. 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)))))))
102		psropprmul.z	. . 3 ⊢ 𝑍 = (𝐼 mPwSer 𝑆)
103		eqid 2729	. . 3 ⊢ (Base‘𝑍) = (Base‘𝑍)
104		psropprmul.u	. . 3 ⊢ ∙ = (.r‘𝑍)
105	90	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (Base‘𝑅) = (Base‘𝑆))
106	105	psrbaspropd 22119	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆)))
107	11	fveq2i 6861	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘(𝐼 mPwSer 𝑅))
108	13, 107	eqtri 2752	. . . . 5 ⊢ 𝐵 = (Base‘(𝐼 mPwSer 𝑅))
109	102	fveq2i 6861	. . . . 5 ⊢ (Base‘𝑍) = (Base‘(𝐼 mPwSer 𝑆))
110	106, 108, 109	3eqtr4g 2789	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐵 = (Base‘𝑍))
111	20, 110	eleqtrd 2830	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ (Base‘𝑍))
112	14, 110	eleqtrd 2830	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ (Base‘𝑍))
113	102, 103, 79, 104, 12, 111, 112	psrmulfval 21852	. 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)))))))
114	99, 101, 113	3eqtr4rd 2775	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) = (𝐺 · 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {crab 3405  Vcvv 3447   ⊆ wss 3914   class class class wbr 5107   ↦ cmpt 5188  ^◡ccnv 5637  dom cdm 5638   “ cima 5641   ∘ ccom 5642  Fun wfun 6505  ⟶wf 6507  –1-1-onto→wf1o 6510  ‘cfv 6511  (class class class)co 7387   ∘_f cof 7651   ∘_r cofr 7652   supp csupp 8139   ↑_m cmap 8799  Fincfn 8918   finSupp cfsupp 9312  ℂcc 11066   ≤ cle 11209   − cmin 11405  ℕcn 12186  ℕ₀cn0 12442  Basecbs 17179  +_gcplusg 17220  ._rcmulr 17221  0_gc0g 17402   Σ_g cgsu 17403  CMndccmn 19710  Ringcrg 20142  opp_rcoppr 20245   mPwSer cmps 21813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-ofr 7654  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-pm 8802  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fsupp 9313  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-uz 12794  df-fz 13469  df-fzo 13616  df-seq 13967  df-hash 14296  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-tset 17239  df-0g 17404  df-gsum 17405  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-grp 18868  df-minusg 18869  df-cntz 19249  df-cmn 19712  df-abl 19713  df-mgp 20050  df-ur 20091  df-ring 20144  df-oppr 20246  df-psr 21818

This theorem is referenced by:  ply1opprmul  22123