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Theorem rngoisocnv 34105
Description: The inverse of a ring isomorphism is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
Assertion
Ref Expression
rngoisocnv ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑆 RngIso 𝑅))

Proof of Theorem rngoisocnv
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1ocnv 6290 . . . . . . . 8 (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) → 𝐹:ran (1st𝑆)–1-1-onto→ran (1st𝑅))
2 f1of 6278 . . . . . . . 8 (𝐹:ran (1st𝑆)–1-1-onto→ran (1st𝑅) → 𝐹:ran (1st𝑆)⟶ran (1st𝑅))
31, 2syl 17 . . . . . . 7 (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) → 𝐹:ran (1st𝑆)⟶ran (1st𝑅))
43ad2antll 700 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → 𝐹:ran (1st𝑆)⟶ran (1st𝑅))
5 eqid 2770 . . . . . . . . . 10 (2nd𝑅) = (2nd𝑅)
6 eqid 2770 . . . . . . . . . 10 (GId‘(2nd𝑅)) = (GId‘(2nd𝑅))
7 eqid 2770 . . . . . . . . . 10 (2nd𝑆) = (2nd𝑆)
8 eqid 2770 . . . . . . . . . 10 (GId‘(2nd𝑆)) = (GId‘(2nd𝑆))
95, 6, 7, 8rngohom1 34092 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)))
1093expa 1110 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)))
1110adantrr 688 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)))
12 eqid 2770 . . . . . . . . . . 11 ran (1st𝑅) = ran (1st𝑅)
1312, 5, 6rngo1cl 34063 . . . . . . . . . 10 (𝑅 ∈ RingOps → (GId‘(2nd𝑅)) ∈ ran (1st𝑅))
14 f1ocnvfv 6676 . . . . . . . . . 10 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (GId‘(2nd𝑅)) ∈ ran (1st𝑅)) → ((𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)) → (𝐹‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑅))))
1513, 14sylan2 572 . . . . . . . . 9 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ 𝑅 ∈ RingOps) → ((𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)) → (𝐹‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑅))))
1615ancoms 455 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → ((𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)) → (𝐹‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑅))))
1716ad2ant2rl 735 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → ((𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)) → (𝐹‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑅))))
1811, 17mpd 15 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → (𝐹‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑅)))
19 f1ocnvfv2 6675 . . . . . . . . . . . . . 14 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ 𝑥 ∈ ran (1st𝑆)) → (𝐹‘(𝐹𝑥)) = 𝑥)
20 f1ocnvfv2 6675 . . . . . . . . . . . . . 14 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝐹‘(𝐹𝑦)) = 𝑦)
2119, 20anim12da 33831 . . . . . . . . . . . . 13 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹𝑥)) = 𝑥 ∧ (𝐹‘(𝐹𝑦)) = 𝑦))
22 oveq12 6801 . . . . . . . . . . . . 13 (((𝐹‘(𝐹𝑥)) = 𝑥 ∧ (𝐹‘(𝐹𝑦)) = 𝑦) → ((𝐹‘(𝐹𝑥))(1st𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(1st𝑆)𝑦))
2321, 22syl 17 . . . . . . . . . . . 12 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹𝑥))(1st𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(1st𝑆)𝑦))
2423adantll 685 . . . . . . . . . . 11 (((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹𝑥))(1st𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(1st𝑆)𝑦))
2524adantll 685 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹𝑥))(1st𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(1st𝑆)𝑦))
26 f1ocnvdm 6682 . . . . . . . . . . . . . . . 16 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ 𝑥 ∈ ran (1st𝑆)) → (𝐹𝑥) ∈ ran (1st𝑅))
27 f1ocnvdm 6682 . . . . . . . . . . . . . . . 16 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝐹𝑦) ∈ ran (1st𝑅))
2826, 27anim12da 33831 . . . . . . . . . . . . . . 15 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹𝑥) ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ ran (1st𝑅)))
29 eqid 2770 . . . . . . . . . . . . . . . 16 (1st𝑅) = (1st𝑅)
30 eqid 2770 . . . . . . . . . . . . . . . 16 (1st𝑆) = (1st𝑆)
3129, 12, 30rngohomadd 34093 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ ((𝐹𝑥) ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ ran (1st𝑅))) → (𝐹‘((𝐹𝑥)(1st𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(1st𝑆)(𝐹‘(𝐹𝑦))))
3228, 31sylan2 572 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)))) → (𝐹‘((𝐹𝑥)(1st𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(1st𝑆)(𝐹‘(𝐹𝑦))))
3332exp32 407 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) → ((𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝐹‘((𝐹𝑥)(1st𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(1st𝑆)(𝐹‘(𝐹𝑦))))))
34333expa 1110 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) → ((𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝐹‘((𝐹𝑥)(1st𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(1st𝑆)(𝐹‘(𝐹𝑦))))))
3534impr 442 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → ((𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝐹‘((𝐹𝑥)(1st𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(1st𝑆)(𝐹‘(𝐹𝑦)))))
3635imp 393 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘((𝐹𝑥)(1st𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(1st𝑆)(𝐹‘(𝐹𝑦))))
37 eqid 2770 . . . . . . . . . . . . . . . 16 ran (1st𝑆) = ran (1st𝑆)
3830, 37rngogcl 34036 . . . . . . . . . . . . . . 15 ((𝑆 ∈ RingOps ∧ 𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝑥(1st𝑆)𝑦) ∈ ran (1st𝑆))
39383expb 1112 . . . . . . . . . . . . . 14 ((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝑥(1st𝑆)𝑦) ∈ ran (1st𝑆))
40 f1ocnvfv2 6675 . . . . . . . . . . . . . . 15 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥(1st𝑆)𝑦) ∈ ran (1st𝑆)) → (𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝑥(1st𝑆)𝑦))
4140ancoms 455 . . . . . . . . . . . . . 14 (((𝑥(1st𝑆)𝑦) ∈ ran (1st𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝑥(1st𝑆)𝑦))
4239, 41sylan 561 . . . . . . . . . . . . 13 (((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝑥(1st𝑆)𝑦))
4342an32s 623 . . . . . . . . . . . 12 (((𝑆 ∈ RingOps ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝑥(1st𝑆)𝑦))
4443adantlll 689 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝑥(1st𝑆)𝑦))
4544adantlrl 691 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝑥(1st𝑆)𝑦))
4625, 36, 453eqtr4rd 2815 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝐹‘((𝐹𝑥)(1st𝑅)(𝐹𝑦))))
47 f1of1 6277 . . . . . . . . . . . 12 (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) → 𝐹:ran (1st𝑅)–1-1→ran (1st𝑆))
4847ad2antlr 698 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → 𝐹:ran (1st𝑅)–1-1→ran (1st𝑆))
49 f1ocnvdm 6682 . . . . . . . . . . . . . . 15 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥(1st𝑆)𝑦) ∈ ran (1st𝑆)) → (𝐹‘(𝑥(1st𝑆)𝑦)) ∈ ran (1st𝑅))
5049ancoms 455 . . . . . . . . . . . . . 14 (((𝑥(1st𝑆)𝑦) ∈ ran (1st𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝑥(1st𝑆)𝑦)) ∈ ran (1st𝑅))
5139, 50sylan 561 . . . . . . . . . . . . 13 (((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝑥(1st𝑆)𝑦)) ∈ ran (1st𝑅))
5251an32s 623 . . . . . . . . . . . 12 (((𝑆 ∈ RingOps ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝑥(1st𝑆)𝑦)) ∈ ran (1st𝑅))
5352adantlll 689 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝑥(1st𝑆)𝑦)) ∈ ran (1st𝑅))
5429, 12rngogcl 34036 . . . . . . . . . . . . . . 15 ((𝑅 ∈ RingOps ∧ (𝐹𝑥) ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ ran (1st𝑅)) → ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
55543expb 1112 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ ((𝐹𝑥) ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ ran (1st𝑅))) → ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
5628, 55sylan2 572 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)))) → ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
5756anassrs 458 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
5857adantllr 690 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
59 f1fveq 6661 . . . . . . . . . . 11 ((𝐹:ran (1st𝑅)–1-1→ran (1st𝑆) ∧ ((𝐹‘(𝑥(1st𝑆)𝑦)) ∈ ran (1st𝑅) ∧ ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))) → ((𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝐹‘((𝐹𝑥)(1st𝑅)(𝐹𝑦))) ↔ (𝐹‘(𝑥(1st𝑆)𝑦)) = ((𝐹𝑥)(1st𝑅)(𝐹𝑦))))
6048, 53, 58, 59syl12anc 1473 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝐹‘((𝐹𝑥)(1st𝑅)(𝐹𝑦))) ↔ (𝐹‘(𝑥(1st𝑆)𝑦)) = ((𝐹𝑥)(1st𝑅)(𝐹𝑦))))
6160adantlrl 691 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝐹‘((𝐹𝑥)(1st𝑅)(𝐹𝑦))) ↔ (𝐹‘(𝑥(1st𝑆)𝑦)) = ((𝐹𝑥)(1st𝑅)(𝐹𝑦))))
6246, 61mpbid 222 . . . . . . . 8 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝑥(1st𝑆)𝑦)) = ((𝐹𝑥)(1st𝑅)(𝐹𝑦)))
63 oveq12 6801 . . . . . . . . . . . . 13 (((𝐹‘(𝐹𝑥)) = 𝑥 ∧ (𝐹‘(𝐹𝑦)) = 𝑦) → ((𝐹‘(𝐹𝑥))(2nd𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(2nd𝑆)𝑦))
6421, 63syl 17 . . . . . . . . . . . 12 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹𝑥))(2nd𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(2nd𝑆)𝑦))
6564adantll 685 . . . . . . . . . . 11 (((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹𝑥))(2nd𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(2nd𝑆)𝑦))
6665adantll 685 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹𝑥))(2nd𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(2nd𝑆)𝑦))
6729, 12, 5, 7rngohommul 34094 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ ((𝐹𝑥) ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ ran (1st𝑅))) → (𝐹‘((𝐹𝑥)(2nd𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(2nd𝑆)(𝐹‘(𝐹𝑦))))
6828, 67sylan2 572 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)))) → (𝐹‘((𝐹𝑥)(2nd𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(2nd𝑆)(𝐹‘(𝐹𝑦))))
6968exp32 407 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) → ((𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝐹‘((𝐹𝑥)(2nd𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(2nd𝑆)(𝐹‘(𝐹𝑦))))))
70693expa 1110 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) → ((𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝐹‘((𝐹𝑥)(2nd𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(2nd𝑆)(𝐹‘(𝐹𝑦))))))
7170impr 442 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → ((𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝐹‘((𝐹𝑥)(2nd𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(2nd𝑆)(𝐹‘(𝐹𝑦)))))
7271imp 393 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘((𝐹𝑥)(2nd𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(2nd𝑆)(𝐹‘(𝐹𝑦))))
7330, 7, 37rngocl 34025 . . . . . . . . . . . . . . 15 ((𝑆 ∈ RingOps ∧ 𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝑥(2nd𝑆)𝑦) ∈ ran (1st𝑆))
74733expb 1112 . . . . . . . . . . . . . 14 ((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝑥(2nd𝑆)𝑦) ∈ ran (1st𝑆))
75 f1ocnvfv2 6675 . . . . . . . . . . . . . . 15 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥(2nd𝑆)𝑦) ∈ ran (1st𝑆)) → (𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝑥(2nd𝑆)𝑦))
7675ancoms 455 . . . . . . . . . . . . . 14 (((𝑥(2nd𝑆)𝑦) ∈ ran (1st𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝑥(2nd𝑆)𝑦))
7774, 76sylan 561 . . . . . . . . . . . . 13 (((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝑥(2nd𝑆)𝑦))
7877an32s 623 . . . . . . . . . . . 12 (((𝑆 ∈ RingOps ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝑥(2nd𝑆)𝑦))
7978adantlll 689 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝑥(2nd𝑆)𝑦))
8079adantlrl 691 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝑥(2nd𝑆)𝑦))
8166, 72, 803eqtr4rd 2815 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝐹‘((𝐹𝑥)(2nd𝑅)(𝐹𝑦))))
82 f1ocnvdm 6682 . . . . . . . . . . . . . . 15 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥(2nd𝑆)𝑦) ∈ ran (1st𝑆)) → (𝐹‘(𝑥(2nd𝑆)𝑦)) ∈ ran (1st𝑅))
8382ancoms 455 . . . . . . . . . . . . . 14 (((𝑥(2nd𝑆)𝑦) ∈ ran (1st𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝑥(2nd𝑆)𝑦)) ∈ ran (1st𝑅))
8474, 83sylan 561 . . . . . . . . . . . . 13 (((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝑥(2nd𝑆)𝑦)) ∈ ran (1st𝑅))
8584an32s 623 . . . . . . . . . . . 12 (((𝑆 ∈ RingOps ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝑥(2nd𝑆)𝑦)) ∈ ran (1st𝑅))
8685adantlll 689 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝑥(2nd𝑆)𝑦)) ∈ ran (1st𝑅))
8729, 5, 12rngocl 34025 . . . . . . . . . . . . . . 15 ((𝑅 ∈ RingOps ∧ (𝐹𝑥) ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ ran (1st𝑅)) → ((𝐹𝑥)(2nd𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
88873expb 1112 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ ((𝐹𝑥) ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ ran (1st𝑅))) → ((𝐹𝑥)(2nd𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
8928, 88sylan2 572 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)))) → ((𝐹𝑥)(2nd𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
9089anassrs 458 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹𝑥)(2nd𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
9190adantllr 690 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹𝑥)(2nd𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
92 f1fveq 6661 . . . . . . . . . . 11 ((𝐹:ran (1st𝑅)–1-1→ran (1st𝑆) ∧ ((𝐹‘(𝑥(2nd𝑆)𝑦)) ∈ ran (1st𝑅) ∧ ((𝐹𝑥)(2nd𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))) → ((𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝐹‘((𝐹𝑥)(2nd𝑅)(𝐹𝑦))) ↔ (𝐹‘(𝑥(2nd𝑆)𝑦)) = ((𝐹𝑥)(2nd𝑅)(𝐹𝑦))))
9348, 86, 91, 92syl12anc 1473 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝐹‘((𝐹𝑥)(2nd𝑅)(𝐹𝑦))) ↔ (𝐹‘(𝑥(2nd𝑆)𝑦)) = ((𝐹𝑥)(2nd𝑅)(𝐹𝑦))))
9493adantlrl 691 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝐹‘((𝐹𝑥)(2nd𝑅)(𝐹𝑦))) ↔ (𝐹‘(𝑥(2nd𝑆)𝑦)) = ((𝐹𝑥)(2nd𝑅)(𝐹𝑦))))
9581, 94mpbid 222 . . . . . . . 8 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝑥(2nd𝑆)𝑦)) = ((𝐹𝑥)(2nd𝑅)(𝐹𝑦)))
9662, 95jca 495 . . . . . . 7 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝑥(1st𝑆)𝑦)) = ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑆)𝑦)) = ((𝐹𝑥)(2nd𝑅)(𝐹𝑦))))
9796ralrimivva 3119 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → ∀𝑥 ∈ ran (1st𝑆)∀𝑦 ∈ ran (1st𝑆)((𝐹‘(𝑥(1st𝑆)𝑦)) = ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑆)𝑦)) = ((𝐹𝑥)(2nd𝑅)(𝐹𝑦))))
9830, 7, 37, 8, 29, 5, 12, 6isrngohom 34089 . . . . . . . 8 ((𝑆 ∈ RingOps ∧ 𝑅 ∈ RingOps) → (𝐹 ∈ (𝑆 RngHom 𝑅) ↔ (𝐹:ran (1st𝑆)⟶ran (1st𝑅) ∧ (𝐹‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑅)) ∧ ∀𝑥 ∈ ran (1st𝑆)∀𝑦 ∈ ran (1st𝑆)((𝐹‘(𝑥(1st𝑆)𝑦)) = ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑆)𝑦)) = ((𝐹𝑥)(2nd𝑅)(𝐹𝑦))))))
9998ancoms 455 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑆 RngHom 𝑅) ↔ (𝐹:ran (1st𝑆)⟶ran (1st𝑅) ∧ (𝐹‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑅)) ∧ ∀𝑥 ∈ ran (1st𝑆)∀𝑦 ∈ ran (1st𝑆)((𝐹‘(𝑥(1st𝑆)𝑦)) = ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑆)𝑦)) = ((𝐹𝑥)(2nd𝑅)(𝐹𝑦))))))
10099adantr 466 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → (𝐹 ∈ (𝑆 RngHom 𝑅) ↔ (𝐹:ran (1st𝑆)⟶ran (1st𝑅) ∧ (𝐹‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑅)) ∧ ∀𝑥 ∈ ran (1st𝑆)∀𝑦 ∈ ran (1st𝑆)((𝐹‘(𝑥(1st𝑆)𝑦)) = ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑆)𝑦)) = ((𝐹𝑥)(2nd𝑅)(𝐹𝑦))))))
1014, 18, 97, 100mpbir3and 1426 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → 𝐹 ∈ (𝑆 RngHom 𝑅))
1021ad2antll 700 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → 𝐹:ran (1st𝑆)–1-1-onto→ran (1st𝑅))
103101, 102jca 495 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → (𝐹 ∈ (𝑆 RngHom 𝑅) ∧ 𝐹:ran (1st𝑆)–1-1-onto→ran (1st𝑅)))
104103ex 397 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹 ∈ (𝑆 RngHom 𝑅) ∧ 𝐹:ran (1st𝑆)–1-1-onto→ran (1st𝑅))))
10529, 12, 30, 37isrngoiso 34102 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))))
10630, 37, 29, 12isrngoiso 34102 . . . 4 ((𝑆 ∈ RingOps ∧ 𝑅 ∈ RingOps) → (𝐹 ∈ (𝑆 RngIso 𝑅) ↔ (𝐹 ∈ (𝑆 RngHom 𝑅) ∧ 𝐹:ran (1st𝑆)–1-1-onto→ran (1st𝑅))))
107106ancoms 455 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑆 RngIso 𝑅) ↔ (𝐹 ∈ (𝑆 RngHom 𝑅) ∧ 𝐹:ran (1st𝑆)–1-1-onto→ran (1st𝑅))))
108104, 105, 1073imtr4d 283 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝐹 ∈ (𝑆 RngIso 𝑅)))
1091083impia 1108 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑆 RngIso 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wral 3060  ccnv 5248  ran crn 5250  wf 6027  1-1wf1 6028  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6792  1st c1st 7312  2nd c2nd 7313  GIdcgi 27678  RingOpscrngo 34018   RngHom crnghom 34084   RngIso crngiso 34085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-map 8010  df-grpo 27681  df-gid 27682  df-ablo 27733  df-ass 33967  df-exid 33969  df-mgmOLD 33973  df-sgrOLD 33985  df-mndo 33991  df-rngo 34019  df-rngohom 34087  df-rngoiso 34100
This theorem is referenced by:  riscer  34112
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