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Theorem rngoisocnv 33433
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 6108 . . . . . . . 8 (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) → 𝐹:ran (1st𝑆)–1-1-onto→ran (1st𝑅))
2 f1of 6096 . . . . . . . 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 764 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → 𝐹:ran (1st𝑆)⟶ran (1st𝑅))
5 eqid 2621 . . . . . . . . . 10 (2nd𝑅) = (2nd𝑅)
6 eqid 2621 . . . . . . . . . 10 (GId‘(2nd𝑅)) = (GId‘(2nd𝑅))
7 eqid 2621 . . . . . . . . . 10 (2nd𝑆) = (2nd𝑆)
8 eqid 2621 . . . . . . . . . 10 (GId‘(2nd𝑆)) = (GId‘(2nd𝑆))
95, 6, 7, 8rngohom1 33420 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)))
1093expa 1262 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)))
1110adantrr 752 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)))
12 eqid 2621 . . . . . . . . . . 11 ran (1st𝑅) = ran (1st𝑅)
1312, 5, 6rngo1cl 33391 . . . . . . . . . 10 (𝑅 ∈ RingOps → (GId‘(2nd𝑅)) ∈ ran (1st𝑅))
14 f1ocnvfv 6491 . . . . . . . . . 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 491 . . . . . . . . 9 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ 𝑅 ∈ RingOps) → ((𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)) → (𝐹‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑅))))
1615ancoms 469 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → ((𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)) → (𝐹‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑅))))
1716ad2ant2rl 784 . . . . . . 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 6490 . . . . . . . . . . . . . 14 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ 𝑥 ∈ ran (1st𝑆)) → (𝐹‘(𝐹𝑥)) = 𝑥)
20 f1ocnvfv2 6490 . . . . . . . . . . . . . 14 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝐹‘(𝐹𝑦)) = 𝑦)
2119, 20anim12da 33159 . . . . . . . . . . . . 13 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹𝑥)) = 𝑥 ∧ (𝐹‘(𝐹𝑦)) = 𝑦))
22 oveq12 6616 . . . . . . . . . . . . 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 749 . . . . . . . . . . 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 749 . . . . . . . . . 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 6497 . . . . . . . . . . . . . . . 16 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ 𝑥 ∈ ran (1st𝑆)) → (𝐹𝑥) ∈ ran (1st𝑅))
27 f1ocnvdm 6497 . . . . . . . . . . . . . . . 16 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝐹𝑦) ∈ ran (1st𝑅))
2826, 27anim12da 33159 . . . . . . . . . . . . . . 15 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹𝑥) ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ ran (1st𝑅)))
29 eqid 2621 . . . . . . . . . . . . . . . 16 (1st𝑅) = (1st𝑅)
30 eqid 2621 . . . . . . . . . . . . . . . 16 (1st𝑆) = (1st𝑆)
3129, 12, 30rngohomadd 33421 . . . . . . . . . . . . . . 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 491 . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . 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 1262 . . . . . . . . . . . 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 648 . . . . . . . . . . 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 445 . . . . . . . . . 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 2621 . . . . . . . . . . . . . . . 16 ran (1st𝑆) = ran (1st𝑆)
3830, 37rngogcl 33364 . . . . . . . . . . . . . . 15 ((𝑆 ∈ RingOps ∧ 𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝑥(1st𝑆)𝑦) ∈ ran (1st𝑆))
39383expb 1263 . . . . . . . . . . . . . 14 ((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝑥(1st𝑆)𝑦) ∈ ran (1st𝑆))
40 f1ocnvfv2 6490 . . . . . . . . . . . . . . 15 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥(1st𝑆)𝑦) ∈ ran (1st𝑆)) → (𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝑥(1st𝑆)𝑦))
4140ancoms 469 . . . . . . . . . . . . . 14 (((𝑥(1st𝑆)𝑦) ∈ ran (1st𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝑥(1st𝑆)𝑦))
4239, 41sylan 488 . . . . . . . . . . . . 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 845 . . . . . . . . . . . 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 753 . . . . . . . . . . 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 755 . . . . . . . . . 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 2666 . . . . . . . . 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 6095 . . . . . . . . . . . 12 (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) → 𝐹:ran (1st𝑅)–1-1→ran (1st𝑆))
4847ad2antlr 762 . . . . . . . . . . 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 6497 . . . . . . . . . . . . . . 15 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥(1st𝑆)𝑦) ∈ ran (1st𝑆)) → (𝐹‘(𝑥(1st𝑆)𝑦)) ∈ ran (1st𝑅))
5049ancoms 469 . . . . . . . . . . . . . 14 (((𝑥(1st𝑆)𝑦) ∈ ran (1st𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝑥(1st𝑆)𝑦)) ∈ ran (1st𝑅))
5139, 50sylan 488 . . . . . . . . . . . . 13 (((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝑥(1st𝑆)𝑦)) ∈ ran (1st𝑅))
5251an32s 845 . . . . . . . . . . . 12 (((𝑆 ∈ RingOps ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝑥(1st𝑆)𝑦)) ∈ ran (1st𝑅))
5352adantlll 753 . . . . . . . . . . 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 33364 . . . . . . . . . . . . . . 15 ((𝑅 ∈ RingOps ∧ (𝐹𝑥) ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ ran (1st𝑅)) → ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
55543expb 1263 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ ((𝐹𝑥) ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ ran (1st𝑅))) → ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
5628, 55sylan2 491 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)))) → ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
5756anassrs 679 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
5857adantllr 754 . . . . . . . . . . 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 6476 . . . . . . . . . . 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 1321 . . . . . . . . . 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 755 . . . . . . . . 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 6616 . . . . . . . . . . . . 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 749 . . . . . . . . . . 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 749 . . . . . . . . . 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 33422 . . . . . . . . . . . . . . 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 491 . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . 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 1262 . . . . . . . . . . . 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 648 . . . . . . . . . . 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 445 . . . . . . . . . 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 33353 . . . . . . . . . . . . . . 15 ((𝑆 ∈ RingOps ∧ 𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝑥(2nd𝑆)𝑦) ∈ ran (1st𝑆))
74733expb 1263 . . . . . . . . . . . . . 14 ((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝑥(2nd𝑆)𝑦) ∈ ran (1st𝑆))
75 f1ocnvfv2 6490 . . . . . . . . . . . . . . 15 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥(2nd𝑆)𝑦) ∈ ran (1st𝑆)) → (𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝑥(2nd𝑆)𝑦))
7675ancoms 469 . . . . . . . . . . . . . 14 (((𝑥(2nd𝑆)𝑦) ∈ ran (1st𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝑥(2nd𝑆)𝑦))
7774, 76sylan 488 . . . . . . . . . . . . 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 845 . . . . . . . . . . . 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 753 . . . . . . . . . . 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 755 . . . . . . . . . 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 2666 . . . . . . . . 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 6497 . . . . . . . . . . . . . . 15 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥(2nd𝑆)𝑦) ∈ ran (1st𝑆)) → (𝐹‘(𝑥(2nd𝑆)𝑦)) ∈ ran (1st𝑅))
8382ancoms 469 . . . . . . . . . . . . . 14 (((𝑥(2nd𝑆)𝑦) ∈ ran (1st𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝑥(2nd𝑆)𝑦)) ∈ ran (1st𝑅))
8474, 83sylan 488 . . . . . . . . . . . . 13 (((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝑥(2nd𝑆)𝑦)) ∈ ran (1st𝑅))
8584an32s 845 . . . . . . . . . . . 12 (((𝑆 ∈ RingOps ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝑥(2nd𝑆)𝑦)) ∈ ran (1st𝑅))
8685adantlll 753 . . . . . . . . . . 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 33353 . . . . . . . . . . . . . . 15 ((𝑅 ∈ RingOps ∧ (𝐹𝑥) ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ ran (1st𝑅)) → ((𝐹𝑥)(2nd𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
88873expb 1263 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ ((𝐹𝑥) ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ ran (1st𝑅))) → ((𝐹𝑥)(2nd𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
8928, 88sylan2 491 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)))) → ((𝐹𝑥)(2nd𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
9089anassrs 679 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹𝑥)(2nd𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
9190adantllr 754 . . . . . . . . . . 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 6476 . . . . . . . . . . 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 1321 . . . . . . . . . 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 755 . . . . . . . . 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 554 . . . . . . 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 2965 . . . . . 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 33417 . . . . . . . 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 469 . . . . . . 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 481 . . . . . 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 1243 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → 𝐹 ∈ (𝑆 RngHom 𝑅))
1021ad2antll 764 . . . . 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 554 . . . 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 450 . . 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 33430 . . 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 33430 . . . 4 ((𝑆 ∈ RingOps ∧ 𝑅 ∈ RingOps) → (𝐹 ∈ (𝑆 RngIso 𝑅) ↔ (𝐹 ∈ (𝑆 RngHom 𝑅) ∧ 𝐹:ran (1st𝑆)–1-1-onto→ran (1st𝑅))))
107106ancoms 469 . . 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 1258 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  ccnv 5075  ran crn 5077  wf 5845  1-1wf1 5846  1-1-ontowf1o 5848  cfv 5849  (class class class)co 6607  1st c1st 7114  2nd c2nd 7115  GIdcgi 27205  RingOpscrngo 33346   RngHom crnghom 33412   RngIso crngiso 33413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-1st 7116  df-2nd 7117  df-map 7807  df-grpo 27208  df-gid 27209  df-ablo 27260  df-ass 33295  df-exid 33297  df-mgmOLD 33301  df-sgrOLD 33313  df-mndo 33319  df-rngo 33347  df-rngohom 33415  df-rngoiso 33428
This theorem is referenced by:  riscer  33440
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