Theorem psropprmul 20406

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 2821	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2821	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
3		ringcmn 19331	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
4	3	3ad2ant1 1129	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝑅 ∈ CMnd)
5	4	adantr 483	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → 𝑅 ∈ CMnd)
6		ovex 7189	. . . . . . . 8 ⊢ (ℕ0 ↑m 𝐼) ∈ V
7	6	rabex 5235	. . . . . . 7 ⊢ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∈ V
8	7	rabex 5235	. . . . . 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 1208	. . . . . . 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 2821	. . . . . . . . . 10 ⊢ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} = {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}
13		psropprmul.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑌)
14		simp3 1134	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵)
15	11, 1, 12, 13, 14	psrelbas 20159	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺:{𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
16	15	adantr 483	. . . . . . . 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 3675	. . . . . . . 8 ⊢ (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘r ≤ 𝑏} → 𝑒 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin})
18		ffvelrn 6849	. . . . . . . 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 1133	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵)
21	11, 1, 12, 13, 20	psrelbas 20159	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹:{𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
22	21	ad2antrr 724	. . . . . . . 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 4056	. . . . . . . . 9 ⊢ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘r ≤ 𝑏} ⊆ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}
24		reldmpsr 20141	. . . . . . . . . . . . 13 ⊢ Rel dom mPwSer
25	11, 13, 24	strov2rcl 16546	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ 𝐵 → 𝐼 ∈ V)
26	25	3ad2ant3 1131	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐼 ∈ V)
27	26	ad2antrr 724	. . . . . . . . . 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 767	. . . . . . . . . 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 487	. . . . . . . . . 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 2821	. . . . . . . . . . 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 20153	. . . . . . . . . 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 1367	. . . . . . . . 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 3965	. . . . . . . 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 6852	. . . . . . 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 2821	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
36	1, 35	ringcl 19311	. . . . . . 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 1367	. . . . . 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	37	fmpttd 6879	. . . . 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))
39		mptexg 6984	. . . . . . 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)
40	8, 39	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)
41		funmpt 6393	. . . . . . 7 ⊢ Fun (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘r ≤ 𝑏} ↦ ((𝐺‘𝑒)(.r‘𝑅)(𝐹‘(𝑏 ∘f − 𝑒))))
42	41	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)))))
43		fvexd 6685	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → (0g‘𝑅) ∈ V)
44	12	psrbaglefi 20152	. . . . . . 7 ⊢ ((𝐼 ∈ V ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘r ≤ 𝑏} ∈ Fin)
45	26, 44	sylan 582	. . . . . 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)
46		suppssdm 7843	. . . . . . . 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))))
47		eqid 2821	. . . . . . . . 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))))
48	47	dmmptss 6095	. . . . . . . 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}
49	46, 48	sstri 3976	. . . . . . 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}
50	49	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})
51		suppssfifsupp 8848	. . . . . 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))
52	40, 42, 43, 45, 50, 51	syl32anc 1374	. . . . 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))
53	12, 30	psrbagconf1o 20154	. . . . . 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})
54	26, 53	sylan 582	. . . . 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})
55	1, 2, 5, 9, 38, 52, 54	gsumf1o 19036	. . . 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)))))
56	26	ad2antrr 724	. . . . . . . 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)
57		simplr 767	. . . . . . . 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})
58		simpr 487	. . . . . . . 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})
59	12, 30	psrbagconcl 20153	. . . . . . . 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})
60	56, 57, 58, 59	syl3anc 1367	. . . . . . 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})
61		eqidd 2822	. . . . . . 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)))
62		eqidd 2822	. . . . . . 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)))))
63		fveq2 6670	. . . . . . . 8 ⊢ (𝑒 = (𝑏 ∘f − 𝑐) → (𝐺‘𝑒) = (𝐺‘(𝑏 ∘f − 𝑐)))
64		oveq2 7164	. . . . . . . . 9 ⊢ (𝑒 = (𝑏 ∘f − 𝑐) → (𝑏 ∘f − 𝑒) = (𝑏 ∘f − (𝑏 ∘f − 𝑐)))
65	64	fveq2d 6674	. . . . . . . 8 ⊢ (𝑒 = (𝑏 ∘f − 𝑐) → (𝐹‘(𝑏 ∘f − 𝑒)) = (𝐹‘(𝑏 ∘f − (𝑏 ∘f − 𝑐))))
66	63, 65	oveq12d 7174	. . . . . . 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)))))
67	60, 61, 62, 66	fmptco 6891	. . . . . 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))))))
68	12	psrbagf 20145	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ V ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → 𝑏:𝐼⟶ℕ0)
69	26, 68	sylan 582	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → 𝑏:𝐼⟶ℕ0)
70	69	adantr 483	. . . . . . . . . . 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)
71	26	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → 𝐼 ∈ V)
72		elrabi 3675	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘r ≤ 𝑏} → 𝑐 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin})
73	12	psrbagf 20145	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ V ∧ 𝑐 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin}) → 𝑐:𝐼⟶ℕ0)
74	71, 72, 73	syl2an 597	. . . . . . . . . . 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)
75		nn0cn 11908	. . . . . . . . . . . . 13 ⊢ (𝑒 ∈ ℕ0 → 𝑒 ∈ ℂ)
76		nn0cn 11908	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ℕ0 → 𝑓 ∈ ℂ)
77		nncan 10915	. . . . . . . . . . . . 13 ⊢ ((𝑒 ∈ ℂ ∧ 𝑓 ∈ ℂ) → (𝑒 − (𝑒 − 𝑓)) = 𝑓)
78	75, 76, 77	syl2an 597	. . . . . . . . . . . 12 ⊢ ((𝑒 ∈ ℕ0 ∧ 𝑓 ∈ ℕ0) → (𝑒 − (𝑒 − 𝑓)) = 𝑓)
79	78	adantl 484	. . . . . . . . . . 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)
80	56, 70, 74, 79	caonncan 7447	. . . . . . . . . 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)
81	80	fveq2d 6674	. . . . . . . . 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))
82	81	oveq2d 7172	. . . . . . . 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)))
83		psropprmul.s	. . . . . . . . 9 ⊢ 𝑆 = (oppr‘𝑅)
84		eqid 2821	. . . . . . . . 9 ⊢ (.r‘𝑆) = (.r‘𝑆)
85	1, 35, 83, 84	opprmul 19376	. . . . . . . 8 ⊢ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘f − 𝑐))) = ((𝐺‘(𝑏 ∘f − 𝑐))(.r‘𝑅)(𝐹‘𝑐))
86	82, 85	syl6eqr 2874	. . . . . . 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))))
87	86	mpteq2dva 5161	. . . . . 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)))))
88	67, 87	eqtrd 2856	. . . . 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)))))
89	88	oveq2d 7172	. . . 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))))))
90	8	mptex 6986	. . . . . . . 8 ⊢ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘r ≤ 𝑏} ↦ ((𝐹‘𝑐)(.r‘𝑆)(𝐺‘(𝑏 ∘f − 𝑐)))) ∈ V
91	90	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)
92		id 22	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Ring)
93	83	fvexi 6684	. . . . . . . 8 ⊢ 𝑆 ∈ V
94	93	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑆 ∈ V)
95	83, 1	opprbas 19379	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑆)
96	95	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑆))
97		eqid 2821	. . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅)
98	83, 97	oppradd 19380	. . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑆)
99	98	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (+g‘𝑅) = (+g‘𝑆))
100	91, 92, 94, 96, 99	gsumpropd 17888	. . . . . 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))))))
101	100	3ad2ant1 1129	. . . . 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))))))
102	101	adantr 483	. . . 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))))))
103	55, 89, 102	3eqtrd 2860	. . 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))))))
104	103	mpteq2dva 5161	. 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)))))))
105		psropprmul.t	. . 3 ⊢ · = (.r‘𝑌)
106	11, 13, 35, 105, 12, 14, 20	psrmulfval 20165	. 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)))))))
107		psropprmul.z	. . 3 ⊢ 𝑍 = (𝐼 mPwSer 𝑆)
108		eqid 2821	. . 3 ⊢ (Base‘𝑍) = (Base‘𝑍)
109		psropprmul.u	. . 3 ⊢ ∙ = (.r‘𝑍)
110	95	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (Base‘𝑅) = (Base‘𝑆))
111	110	psrbaspropd 20403	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆)))
112	11	fveq2i 6673	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘(𝐼 mPwSer 𝑅))
113	13, 112	eqtri 2844	. . . . 5 ⊢ 𝐵 = (Base‘(𝐼 mPwSer 𝑅))
114	107	fveq2i 6673	. . . . 5 ⊢ (Base‘𝑍) = (Base‘(𝐼 mPwSer 𝑆))
115	111, 113, 114	3eqtr4g 2881	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐵 = (Base‘𝑍))
116	20, 115	eleqtrd 2915	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ (Base‘𝑍))
117	14, 115	eleqtrd 2915	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ (Base‘𝑍))
118	107, 108, 84, 109, 12, 116, 117	psrmulfval 20165	. 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)))))))
119	104, 106, 118	3eqtr4rd 2867	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) = (𝐺 · 𝐹))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  {crab 3142  Vcvv 3494   ⊆ wss 3936   class class class wbr 5066   ↦ cmpt 5146  ^◡ccnv 5554  dom cdm 5555   “ cima 5558   ∘ ccom 5559  Fun wfun 6349  ⟶wf 6351  –1-1-onto→wf1o 6354  ‘cfv 6355  (class class class)co 7156   ∘_f cof 7407   ∘_r cofr 7408   supp csupp 7830   ↑_m cmap 8406  Fincfn 8509   finSupp cfsupp 8833  ℂcc 10535   ≤ cle 10676   − cmin 10870  ℕcn 11638  ℕ₀cn0 11898  Basecbs 16483  +_gcplusg 16565  ._rcmulr 16566  0_gc0g 16713   Σ_g cgsu 16714  CMndccmn 18906  Ringcrg 19297  opp_rcoppr 19372   mPwSer cmps 20131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7409  df-ofr 7410  df-om 7581  df-1st 7689  df-2nd 7690  df-supp 7831  df-tpos 7892  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-map 8408  df-pm 8409  df-ixp 8462  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-fsupp 8834  df-oi 8974  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-uz 12245  df-fz 12894  df-fzo 13035  df-seq 13371  df-hash 13692  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-plusg 16578  df-mulr 16579  df-sca 16581  df-vsca 16582  df-tset 16584  df-0g 16715  df-gsum 16716  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-minusg 18107  df-cntz 18447  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-ring 19299  df-oppr 19373  df-psr 20136

This theorem is referenced by:  ply1opprmul  20407