Theorem psropprmul 22201

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 2736	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2736	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
3		ringcmn 20263	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
4	3	3ad2ant1 1134	. . . . . 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 7400	. . . . . . . 8 ⊢ (ℕ0 ↑m 𝐼) ∈ V
7	6	rabex 5280	. . . . . . 7 ⊢ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∈ V
8	7	rabex 5280	. . . . . 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 1214	. . . . . . 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 2736	. . . . . . . . . 10 ⊢ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} = {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}
13		psropprmul.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑌)
14		simp3 1139	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵)
15	11, 1, 12, 13, 14	psrelbas 21914	. . . . . . . . 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 3630	. . . . . . . 8 ⊢ (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘r ≤ 𝑏} → 𝑒 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin})
18		ffvelcdm 7033	. . . . . . . 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 597	. . . . . . 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 1138	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵)
21	11, 1, 12, 13, 20	psrelbas 21914	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹:{𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
22	21	ad2antrr 727	. . . . . . . 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 4020	. . . . . . . . 9 ⊢ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘r ≤ 𝑏} ⊆ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}
24		eqid 2736	. . . . . . . . . . 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 21907	. . . . . . . . . 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 715	. . . . . . . . 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 3919	. . . . . . . 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 7037	. . . . . . 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 2736	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
30	1, 29	ringcl 20231	. . . . . . 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 1374	. . . . . 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 7067	. . . . 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 7176	. . . . . . 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 6536	. . . . . . 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 6855	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → (0g‘𝑅) ∈ V)
38	12	psrbaglefi 21906	. . . . . . 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 8127	. . . . . . . 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 2736	. . . . . . . . 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 6205	. . . . . . . 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 3931	. . . . . . 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 9293	. . . . . 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 1381	. . . . 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 21909	. . . . . 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 19891	. . . 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 21907	. . . . . . . 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 715	. . . . . . 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 2737	. . . . . . 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 2737	. . . . . . 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 6840	. . . . . . . 8 ⊢ (𝑒 = (𝑏 ∘f − 𝑐) → (𝐺‘𝑒) = (𝐺‘(𝑏 ∘f − 𝑐)))
55		oveq2 7375	. . . . . . . . 9 ⊢ (𝑒 = (𝑏 ∘f − 𝑐) → (𝑏 ∘f − 𝑒) = (𝑏 ∘f − (𝑏 ∘f − 𝑐)))
56	55	fveq2d 6844	. . . . . . . 8 ⊢ (𝑒 = (𝑏 ∘f − 𝑐) → (𝐹‘(𝑏 ∘f − 𝑒)) = (𝐹‘(𝑏 ∘f − (𝑏 ∘f − 𝑐))))
57	54, 56	oveq12d 7385	. . . . . . 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 7082	. . . . . 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 21894	. . . . . . . . . . . . . 14 ⊢ Rel dom mPwSer
60	11, 13, 59	strov2rcl 17187	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ 𝐵 → 𝐼 ∈ V)
61	60	3ad2ant3 1136	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐼 ∈ V)
62	61	ad2antrr 727	. . . . . . . . . . 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 21898	. . . . . . . . . . . . 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 3630	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘r ≤ 𝑏} → 𝑐 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin})
67	12	psrbagf 21898	. . . . . . . . . . . . 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 12447	. . . . . . . . . . . . 13 ⊢ (𝑒 ∈ ℕ0 → 𝑒 ∈ ℂ)
71		nn0cn 12447	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ℕ0 → 𝑓 ∈ ℂ)
72		nncan 11423	. . . . . . . . . . . . 13 ⊢ ((𝑒 ∈ ℂ ∧ 𝑓 ∈ ℂ) → (𝑒 − (𝑒 − 𝑓)) = 𝑓)
73	70, 71, 72	syl2an 597	. . . . . . . . . . . 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 7675	. . . . . . . . . 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 6844	. . . . . . . . 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 7383	. . . . . . . 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 2736	. . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆)
80	1, 29, 78, 79	opprmul 20320	. . . . . . . 8 ⊢ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘f − 𝑐))) = ((𝐺‘(𝑏 ∘f − 𝑐))(.r‘𝑅)(𝐹‘𝑐))
81	77, 80	eqtr4di 2789	. . . . . . 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 5178	. . . . . 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 2771	. . . . 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 7383	. . . 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 7178	. . . . . . . 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 6854	. . . . . . . 8 ⊢ 𝑆 ∈ V
89	88	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑆 ∈ V)
90	78, 1	opprbas 20323	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑆)
91	90	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑆))
92		eqid 2736	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅)
93	78, 92	oppradd 20324	. . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑆)
94	93	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (+g‘𝑅) = (+g‘𝑆))
95	86, 87, 89, 91, 94	gsumpropd 18646	. . . . . 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 1134	. . . . 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 2775	. . 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 5178	. 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 21922	. 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 2736	. . 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 22198	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆)))
107	11	fveq2i 6843	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘(𝐼 mPwSer 𝑅))
108	13, 107	eqtri 2759	. . . . 5 ⊢ 𝐵 = (Base‘(𝐼 mPwSer 𝑅))
109	102	fveq2i 6843	. . . . 5 ⊢ (Base‘𝑍) = (Base‘(𝐼 mPwSer 𝑆))
110	106, 108, 109	3eqtr4g 2796	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐵 = (Base‘𝑍))
111	20, 110	eleqtrd 2838	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ (Base‘𝑍))
112	14, 110	eleqtrd 2838	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ (Base‘𝑍))
113	102, 103, 79, 104, 12, 111, 112	psrmulfval 21922	. 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 2782	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) = (𝐺 · 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {crab 3389  Vcvv 3429   ⊆ wss 3889   class class class wbr 5085   ↦ cmpt 5166  ^◡ccnv 5630  dom cdm 5631   “ cima 5634   ∘ ccom 5635  Fun wfun 6492  ⟶wf 6494  –1-1-onto→wf1o 6497  ‘cfv 6498  (class class class)co 7367   ∘_f cof 7629   ∘_r cofr 7630   supp csupp 8110   ↑_m cmap 8773  Fincfn 8893   finSupp cfsupp 9274  ℂcc 11036   ≤ cle 11180   − cmin 11377  ℕcn 12174  ℕ₀cn0 12437  Basecbs 17179  +_gcplusg 17220  ._rcmulr 17221  0_gc0g 17402   Σ_g cgsu 17403  CMndccmn 19755  Ringcrg 20214  opp_rcoppr 20316   mPwSer cmps 21884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-ofr 7632  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-tpos 8176  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-map 8775  df-pm 8776  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-uz 12789  df-fz 13462  df-fzo 13609  df-seq 13964  df-hash 14293  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 18608  df-sgrp 18687  df-mnd 18703  df-grp 18912  df-minusg 18913  df-cntz 19292  df-cmn 19757  df-abl 19758  df-mgp 20122  df-ur 20163  df-ring 20216  df-oppr 20317  df-psr 21889

This theorem is referenced by:  ply1opprmul  22202