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Theorem rngoisocnv 36443
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)) β†’ ◑𝐹 ∈ (𝑆 RngIso 𝑅))

Proof of Theorem rngoisocnv
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1ocnv 6797 . . . . . . . 8 (𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) β†’ ◑𝐹:ran (1st β€˜π‘†)–1-1-ontoβ†’ran (1st β€˜π‘…))
2 f1of 6785 . . . . . . . 8 (◑𝐹:ran (1st β€˜π‘†)–1-1-ontoβ†’ran (1st β€˜π‘…) β†’ ◑𝐹:ran (1st β€˜π‘†)⟢ran (1st β€˜π‘…))
31, 2syl 17 . . . . . . 7 (𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) β†’ ◑𝐹:ran (1st β€˜π‘†)⟢ran (1st β€˜π‘…))
43ad2antll 728 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) β†’ ◑𝐹:ran (1st β€˜π‘†)⟢ran (1st β€˜π‘…))
5 eqid 2737 . . . . . . . . . 10 (2nd β€˜π‘…) = (2nd β€˜π‘…)
6 eqid 2737 . . . . . . . . . 10 (GIdβ€˜(2nd β€˜π‘…)) = (GIdβ€˜(2nd β€˜π‘…))
7 eqid 2737 . . . . . . . . . 10 (2nd β€˜π‘†) = (2nd β€˜π‘†)
8 eqid 2737 . . . . . . . . . 10 (GIdβ€˜(2nd β€˜π‘†)) = (GIdβ€˜(2nd β€˜π‘†))
95, 6, 7, 8rngohom1 36430 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (πΉβ€˜(GIdβ€˜(2nd β€˜π‘…))) = (GIdβ€˜(2nd β€˜π‘†)))
1093expa 1119 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (πΉβ€˜(GIdβ€˜(2nd β€˜π‘…))) = (GIdβ€˜(2nd β€˜π‘†)))
1110adantrr 716 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) β†’ (πΉβ€˜(GIdβ€˜(2nd β€˜π‘…))) = (GIdβ€˜(2nd β€˜π‘†)))
12 eqid 2737 . . . . . . . . . . 11 ran (1st β€˜π‘…) = ran (1st β€˜π‘…)
1312, 5, 6rngo1cl 36401 . . . . . . . . . 10 (𝑅 ∈ RingOps β†’ (GIdβ€˜(2nd β€˜π‘…)) ∈ ran (1st β€˜π‘…))
14 f1ocnvfv 7225 . . . . . . . . . 10 ((𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) ∧ (GIdβ€˜(2nd β€˜π‘…)) ∈ ran (1st β€˜π‘…)) β†’ ((πΉβ€˜(GIdβ€˜(2nd β€˜π‘…))) = (GIdβ€˜(2nd β€˜π‘†)) β†’ (β—‘πΉβ€˜(GIdβ€˜(2nd β€˜π‘†))) = (GIdβ€˜(2nd β€˜π‘…))))
1513, 14sylan2 594 . . . . . . . . 9 ((𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) ∧ 𝑅 ∈ RingOps) β†’ ((πΉβ€˜(GIdβ€˜(2nd β€˜π‘…))) = (GIdβ€˜(2nd β€˜π‘†)) β†’ (β—‘πΉβ€˜(GIdβ€˜(2nd β€˜π‘†))) = (GIdβ€˜(2nd β€˜π‘…))))
1615ancoms 460 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) β†’ ((πΉβ€˜(GIdβ€˜(2nd β€˜π‘…))) = (GIdβ€˜(2nd β€˜π‘†)) β†’ (β—‘πΉβ€˜(GIdβ€˜(2nd β€˜π‘†))) = (GIdβ€˜(2nd β€˜π‘…))))
1716ad2ant2rl 748 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) β†’ ((πΉβ€˜(GIdβ€˜(2nd β€˜π‘…))) = (GIdβ€˜(2nd β€˜π‘†)) β†’ (β—‘πΉβ€˜(GIdβ€˜(2nd β€˜π‘†))) = (GIdβ€˜(2nd β€˜π‘…))))
1811, 17mpd 15 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) β†’ (β—‘πΉβ€˜(GIdβ€˜(2nd β€˜π‘†))) = (GIdβ€˜(2nd β€˜π‘…)))
19 f1ocnvfv2 7224 . . . . . . . . . . . . . 14 ((𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) ∧ π‘₯ ∈ ran (1st β€˜π‘†)) β†’ (πΉβ€˜(β—‘πΉβ€˜π‘₯)) = π‘₯)
20 f1ocnvfv2 7224 . . . . . . . . . . . . . 14 ((𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†)) β†’ (πΉβ€˜(β—‘πΉβ€˜π‘¦)) = 𝑦)
2119, 20anim12dan 620 . . . . . . . . . . . . 13 ((𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ ((πΉβ€˜(β—‘πΉβ€˜π‘₯)) = π‘₯ ∧ (πΉβ€˜(β—‘πΉβ€˜π‘¦)) = 𝑦))
22 oveq12 7367 . . . . . . . . . . . . 13 (((πΉβ€˜(β—‘πΉβ€˜π‘₯)) = π‘₯ ∧ (πΉβ€˜(β—‘πΉβ€˜π‘¦)) = 𝑦) β†’ ((πΉβ€˜(β—‘πΉβ€˜π‘₯))(1st β€˜π‘†)(πΉβ€˜(β—‘πΉβ€˜π‘¦))) = (π‘₯(1st β€˜π‘†)𝑦))
2321, 22syl 17 . . . . . . . . . . . 12 ((𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ ((πΉβ€˜(β—‘πΉβ€˜π‘₯))(1st β€˜π‘†)(πΉβ€˜(β—‘πΉβ€˜π‘¦))) = (π‘₯(1st β€˜π‘†)𝑦))
2423adantll 713 . . . . . . . . . . 11 (((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ ((πΉβ€˜(β—‘πΉβ€˜π‘₯))(1st β€˜π‘†)(πΉβ€˜(β—‘πΉβ€˜π‘¦))) = (π‘₯(1st β€˜π‘†)𝑦))
2524adantll 713 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ ((πΉβ€˜(β—‘πΉβ€˜π‘₯))(1st β€˜π‘†)(πΉβ€˜(β—‘πΉβ€˜π‘¦))) = (π‘₯(1st β€˜π‘†)𝑦))
26 f1ocnvdm 7232 . . . . . . . . . . . . . . . 16 ((𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) ∧ π‘₯ ∈ ran (1st β€˜π‘†)) β†’ (β—‘πΉβ€˜π‘₯) ∈ ran (1st β€˜π‘…))
27 f1ocnvdm 7232 . . . . . . . . . . . . . . . 16 ((𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†)) β†’ (β—‘πΉβ€˜π‘¦) ∈ ran (1st β€˜π‘…))
2826, 27anim12dan 620 . . . . . . . . . . . . . . 15 ((𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ ((β—‘πΉβ€˜π‘₯) ∈ ran (1st β€˜π‘…) ∧ (β—‘πΉβ€˜π‘¦) ∈ ran (1st β€˜π‘…)))
29 eqid 2737 . . . . . . . . . . . . . . . 16 (1st β€˜π‘…) = (1st β€˜π‘…)
30 eqid 2737 . . . . . . . . . . . . . . . 16 (1st β€˜π‘†) = (1st β€˜π‘†)
3129, 12, 30rngohomadd 36431 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ ((β—‘πΉβ€˜π‘₯) ∈ ran (1st β€˜π‘…) ∧ (β—‘πΉβ€˜π‘¦) ∈ ran (1st β€˜π‘…))) β†’ (πΉβ€˜((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦))) = ((πΉβ€˜(β—‘πΉβ€˜π‘₯))(1st β€˜π‘†)(πΉβ€˜(β—‘πΉβ€˜π‘¦))))
3228, 31sylan2 594 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†)))) β†’ (πΉβ€˜((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦))) = ((πΉβ€˜(β—‘πΉβ€˜π‘₯))(1st β€˜π‘†)(πΉβ€˜(β—‘πΉβ€˜π‘¦))))
3332exp32 422 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) β†’ ((π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†)) β†’ (πΉβ€˜((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦))) = ((πΉβ€˜(β—‘πΉβ€˜π‘₯))(1st β€˜π‘†)(πΉβ€˜(β—‘πΉβ€˜π‘¦))))))
34333expa 1119 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) β†’ ((π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†)) β†’ (πΉβ€˜((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦))) = ((πΉβ€˜(β—‘πΉβ€˜π‘₯))(1st β€˜π‘†)(πΉβ€˜(β—‘πΉβ€˜π‘¦))))))
3534impr 456 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) β†’ ((π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†)) β†’ (πΉβ€˜((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦))) = ((πΉβ€˜(β—‘πΉβ€˜π‘₯))(1st β€˜π‘†)(πΉβ€˜(β—‘πΉβ€˜π‘¦)))))
3635imp 408 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ (πΉβ€˜((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦))) = ((πΉβ€˜(β—‘πΉβ€˜π‘₯))(1st β€˜π‘†)(πΉβ€˜(β—‘πΉβ€˜π‘¦))))
37 eqid 2737 . . . . . . . . . . . . . . . 16 ran (1st β€˜π‘†) = ran (1st β€˜π‘†)
3830, 37rngogcl 36374 . . . . . . . . . . . . . . 15 ((𝑆 ∈ RingOps ∧ π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†)) β†’ (π‘₯(1st β€˜π‘†)𝑦) ∈ ran (1st β€˜π‘†))
39383expb 1121 . . . . . . . . . . . . . 14 ((𝑆 ∈ RingOps ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ (π‘₯(1st β€˜π‘†)𝑦) ∈ ran (1st β€˜π‘†))
40 f1ocnvfv2 7224 . . . . . . . . . . . . . . 15 ((𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) ∧ (π‘₯(1st β€˜π‘†)𝑦) ∈ ran (1st β€˜π‘†)) β†’ (πΉβ€˜(β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦))) = (π‘₯(1st β€˜π‘†)𝑦))
4140ancoms 460 . . . . . . . . . . . . . 14 (((π‘₯(1st β€˜π‘†)𝑦) ∈ ran (1st β€˜π‘†) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) β†’ (πΉβ€˜(β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦))) = (π‘₯(1st β€˜π‘†)𝑦))
4239, 41sylan 581 . . . . . . . . . . . . 13 (((𝑆 ∈ RingOps ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) β†’ (πΉβ€˜(β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦))) = (π‘₯(1st β€˜π‘†)𝑦))
4342an32s 651 . . . . . . . . . . . 12 (((𝑆 ∈ RingOps ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ (πΉβ€˜(β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦))) = (π‘₯(1st β€˜π‘†)𝑦))
4443adantlll 717 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ (πΉβ€˜(β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦))) = (π‘₯(1st β€˜π‘†)𝑦))
4544adantlrl 719 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ (πΉβ€˜(β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦))) = (π‘₯(1st β€˜π‘†)𝑦))
4625, 36, 453eqtr4rd 2788 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ (πΉβ€˜(β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦))) = (πΉβ€˜((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦))))
47 f1of1 6784 . . . . . . . . . . . 12 (𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) β†’ 𝐹:ran (1st β€˜π‘…)–1-1β†’ran (1st β€˜π‘†))
4847ad2antlr 726 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ 𝐹:ran (1st β€˜π‘…)–1-1β†’ran (1st β€˜π‘†))
49 f1ocnvdm 7232 . . . . . . . . . . . . . . 15 ((𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) ∧ (π‘₯(1st β€˜π‘†)𝑦) ∈ ran (1st β€˜π‘†)) β†’ (β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦)) ∈ ran (1st β€˜π‘…))
5049ancoms 460 . . . . . . . . . . . . . 14 (((π‘₯(1st β€˜π‘†)𝑦) ∈ ran (1st β€˜π‘†) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) β†’ (β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦)) ∈ ran (1st β€˜π‘…))
5139, 50sylan 581 . . . . . . . . . . . . 13 (((𝑆 ∈ RingOps ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) β†’ (β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦)) ∈ ran (1st β€˜π‘…))
5251an32s 651 . . . . . . . . . . . 12 (((𝑆 ∈ RingOps ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ (β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦)) ∈ ran (1st β€˜π‘…))
5352adantlll 717 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ (β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦)) ∈ ran (1st β€˜π‘…))
5429, 12rngogcl 36374 . . . . . . . . . . . . . . 15 ((𝑅 ∈ RingOps ∧ (β—‘πΉβ€˜π‘₯) ∈ ran (1st β€˜π‘…) ∧ (β—‘πΉβ€˜π‘¦) ∈ ran (1st β€˜π‘…)) β†’ ((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦)) ∈ ran (1st β€˜π‘…))
55543expb 1121 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ ((β—‘πΉβ€˜π‘₯) ∈ ran (1st β€˜π‘…) ∧ (β—‘πΉβ€˜π‘¦) ∈ ran (1st β€˜π‘…))) β†’ ((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦)) ∈ ran (1st β€˜π‘…))
5628, 55sylan2 594 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ (𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†)))) β†’ ((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦)) ∈ ran (1st β€˜π‘…))
5756anassrs 469 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ ((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦)) ∈ ran (1st β€˜π‘…))
5857adantllr 718 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ ((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦)) ∈ ran (1st β€˜π‘…))
59 f1fveq 7210 . . . . . . . . . . 11 ((𝐹:ran (1st β€˜π‘…)–1-1β†’ran (1st β€˜π‘†) ∧ ((β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦)) ∈ ran (1st β€˜π‘…) ∧ ((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦)) ∈ ran (1st β€˜π‘…))) β†’ ((πΉβ€˜(β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦))) = (πΉβ€˜((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦))) ↔ (β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦)) = ((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦))))
6048, 53, 58, 59syl12anc 836 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ ((πΉβ€˜(β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦))) = (πΉβ€˜((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦))) ↔ (β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦)) = ((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦))))
6160adantlrl 719 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ ((πΉβ€˜(β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦))) = (πΉβ€˜((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦))) ↔ (β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦)) = ((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦))))
6246, 61mpbid 231 . . . . . . . 8 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ (β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦)) = ((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦)))
63 oveq12 7367 . . . . . . . . . . . . 13 (((πΉβ€˜(β—‘πΉβ€˜π‘₯)) = π‘₯ ∧ (πΉβ€˜(β—‘πΉβ€˜π‘¦)) = 𝑦) β†’ ((πΉβ€˜(β—‘πΉβ€˜π‘₯))(2nd β€˜π‘†)(πΉβ€˜(β—‘πΉβ€˜π‘¦))) = (π‘₯(2nd β€˜π‘†)𝑦))
6421, 63syl 17 . . . . . . . . . . . 12 ((𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ ((πΉβ€˜(β—‘πΉβ€˜π‘₯))(2nd β€˜π‘†)(πΉβ€˜(β—‘πΉβ€˜π‘¦))) = (π‘₯(2nd β€˜π‘†)𝑦))
6564adantll 713 . . . . . . . . . . 11 (((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ ((πΉβ€˜(β—‘πΉβ€˜π‘₯))(2nd β€˜π‘†)(πΉβ€˜(β—‘πΉβ€˜π‘¦))) = (π‘₯(2nd β€˜π‘†)𝑦))
6665adantll 713 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ ((πΉβ€˜(β—‘πΉβ€˜π‘₯))(2nd β€˜π‘†)(πΉβ€˜(β—‘πΉβ€˜π‘¦))) = (π‘₯(2nd β€˜π‘†)𝑦))
6729, 12, 5, 7rngohommul 36432 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ ((β—‘πΉβ€˜π‘₯) ∈ ran (1st β€˜π‘…) ∧ (β—‘πΉβ€˜π‘¦) ∈ ran (1st β€˜π‘…))) β†’ (πΉβ€˜((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦))) = ((πΉβ€˜(β—‘πΉβ€˜π‘₯))(2nd β€˜π‘†)(πΉβ€˜(β—‘πΉβ€˜π‘¦))))
6828, 67sylan2 594 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†)))) β†’ (πΉβ€˜((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦))) = ((πΉβ€˜(β—‘πΉβ€˜π‘₯))(2nd β€˜π‘†)(πΉβ€˜(β—‘πΉβ€˜π‘¦))))
6968exp32 422 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) β†’ ((π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†)) β†’ (πΉβ€˜((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦))) = ((πΉβ€˜(β—‘πΉβ€˜π‘₯))(2nd β€˜π‘†)(πΉβ€˜(β—‘πΉβ€˜π‘¦))))))
70693expa 1119 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) β†’ ((π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†)) β†’ (πΉβ€˜((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦))) = ((πΉβ€˜(β—‘πΉβ€˜π‘₯))(2nd β€˜π‘†)(πΉβ€˜(β—‘πΉβ€˜π‘¦))))))
7170impr 456 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) β†’ ((π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†)) β†’ (πΉβ€˜((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦))) = ((πΉβ€˜(β—‘πΉβ€˜π‘₯))(2nd β€˜π‘†)(πΉβ€˜(β—‘πΉβ€˜π‘¦)))))
7271imp 408 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ (πΉβ€˜((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦))) = ((πΉβ€˜(β—‘πΉβ€˜π‘₯))(2nd β€˜π‘†)(πΉβ€˜(β—‘πΉβ€˜π‘¦))))
7330, 7, 37rngocl 36363 . . . . . . . . . . . . . . 15 ((𝑆 ∈ RingOps ∧ π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†)) β†’ (π‘₯(2nd β€˜π‘†)𝑦) ∈ ran (1st β€˜π‘†))
74733expb 1121 . . . . . . . . . . . . . 14 ((𝑆 ∈ RingOps ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ (π‘₯(2nd β€˜π‘†)𝑦) ∈ ran (1st β€˜π‘†))
75 f1ocnvfv2 7224 . . . . . . . . . . . . . . 15 ((𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) ∧ (π‘₯(2nd β€˜π‘†)𝑦) ∈ ran (1st β€˜π‘†)) β†’ (πΉβ€˜(β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦))) = (π‘₯(2nd β€˜π‘†)𝑦))
7675ancoms 460 . . . . . . . . . . . . . 14 (((π‘₯(2nd β€˜π‘†)𝑦) ∈ ran (1st β€˜π‘†) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) β†’ (πΉβ€˜(β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦))) = (π‘₯(2nd β€˜π‘†)𝑦))
7774, 76sylan 581 . . . . . . . . . . . . 13 (((𝑆 ∈ RingOps ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) β†’ (πΉβ€˜(β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦))) = (π‘₯(2nd β€˜π‘†)𝑦))
7877an32s 651 . . . . . . . . . . . 12 (((𝑆 ∈ RingOps ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ (πΉβ€˜(β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦))) = (π‘₯(2nd β€˜π‘†)𝑦))
7978adantlll 717 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ (πΉβ€˜(β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦))) = (π‘₯(2nd β€˜π‘†)𝑦))
8079adantlrl 719 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ (πΉβ€˜(β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦))) = (π‘₯(2nd β€˜π‘†)𝑦))
8166, 72, 803eqtr4rd 2788 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ (πΉβ€˜(β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦))) = (πΉβ€˜((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦))))
82 f1ocnvdm 7232 . . . . . . . . . . . . . . 15 ((𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) ∧ (π‘₯(2nd β€˜π‘†)𝑦) ∈ ran (1st β€˜π‘†)) β†’ (β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦)) ∈ ran (1st β€˜π‘…))
8382ancoms 460 . . . . . . . . . . . . . 14 (((π‘₯(2nd β€˜π‘†)𝑦) ∈ ran (1st β€˜π‘†) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) β†’ (β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦)) ∈ ran (1st β€˜π‘…))
8474, 83sylan 581 . . . . . . . . . . . . 13 (((𝑆 ∈ RingOps ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) β†’ (β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦)) ∈ ran (1st β€˜π‘…))
8584an32s 651 . . . . . . . . . . . 12 (((𝑆 ∈ RingOps ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ (β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦)) ∈ ran (1st β€˜π‘…))
8685adantlll 717 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ (β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦)) ∈ ran (1st β€˜π‘…))
8729, 5, 12rngocl 36363 . . . . . . . . . . . . . . 15 ((𝑅 ∈ RingOps ∧ (β—‘πΉβ€˜π‘₯) ∈ ran (1st β€˜π‘…) ∧ (β—‘πΉβ€˜π‘¦) ∈ ran (1st β€˜π‘…)) β†’ ((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦)) ∈ ran (1st β€˜π‘…))
88873expb 1121 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ ((β—‘πΉβ€˜π‘₯) ∈ ran (1st β€˜π‘…) ∧ (β—‘πΉβ€˜π‘¦) ∈ ran (1st β€˜π‘…))) β†’ ((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦)) ∈ ran (1st β€˜π‘…))
8928, 88sylan2 594 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ (𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†)))) β†’ ((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦)) ∈ ran (1st β€˜π‘…))
9089anassrs 469 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ ((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦)) ∈ ran (1st β€˜π‘…))
9190adantllr 718 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ ((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦)) ∈ ran (1st β€˜π‘…))
92 f1fveq 7210 . . . . . . . . . . 11 ((𝐹:ran (1st β€˜π‘…)–1-1β†’ran (1st β€˜π‘†) ∧ ((β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦)) ∈ ran (1st β€˜π‘…) ∧ ((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦)) ∈ ran (1st β€˜π‘…))) β†’ ((πΉβ€˜(β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦))) = (πΉβ€˜((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦))) ↔ (β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦)) = ((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦))))
9348, 86, 91, 92syl12anc 836 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ ((πΉβ€˜(β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦))) = (πΉβ€˜((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦))) ↔ (β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦)) = ((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦))))
9493adantlrl 719 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ ((πΉβ€˜(β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦))) = (πΉβ€˜((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦))) ↔ (β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦)) = ((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦))))
9581, 94mpbid 231 . . . . . . . 8 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ (β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦)) = ((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦)))
9662, 95jca 513 . . . . . . 7 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) ∧ (π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†))) β†’ ((β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦)) = ((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦)) ∧ (β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦)) = ((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦))))
9796ralrimivva 3198 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) β†’ βˆ€π‘₯ ∈ ran (1st β€˜π‘†)βˆ€π‘¦ ∈ ran (1st β€˜π‘†)((β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦)) = ((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦)) ∧ (β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦)) = ((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦))))
9830, 7, 37, 8, 29, 5, 12, 6isrngohom 36427 . . . . . . . 8 ((𝑆 ∈ RingOps ∧ 𝑅 ∈ RingOps) β†’ (◑𝐹 ∈ (𝑆 RngHom 𝑅) ↔ (◑𝐹:ran (1st β€˜π‘†)⟢ran (1st β€˜π‘…) ∧ (β—‘πΉβ€˜(GIdβ€˜(2nd β€˜π‘†))) = (GIdβ€˜(2nd β€˜π‘…)) ∧ βˆ€π‘₯ ∈ ran (1st β€˜π‘†)βˆ€π‘¦ ∈ ran (1st β€˜π‘†)((β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦)) = ((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦)) ∧ (β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦)) = ((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦))))))
9998ancoms 460 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) β†’ (◑𝐹 ∈ (𝑆 RngHom 𝑅) ↔ (◑𝐹:ran (1st β€˜π‘†)⟢ran (1st β€˜π‘…) ∧ (β—‘πΉβ€˜(GIdβ€˜(2nd β€˜π‘†))) = (GIdβ€˜(2nd β€˜π‘…)) ∧ βˆ€π‘₯ ∈ ran (1st β€˜π‘†)βˆ€π‘¦ ∈ ran (1st β€˜π‘†)((β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦)) = ((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦)) ∧ (β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦)) = ((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦))))))
10099adantr 482 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) β†’ (◑𝐹 ∈ (𝑆 RngHom 𝑅) ↔ (◑𝐹:ran (1st β€˜π‘†)⟢ran (1st β€˜π‘…) ∧ (β—‘πΉβ€˜(GIdβ€˜(2nd β€˜π‘†))) = (GIdβ€˜(2nd β€˜π‘…)) ∧ βˆ€π‘₯ ∈ ran (1st β€˜π‘†)βˆ€π‘¦ ∈ ran (1st β€˜π‘†)((β—‘πΉβ€˜(π‘₯(1st β€˜π‘†)𝑦)) = ((β—‘πΉβ€˜π‘₯)(1st β€˜π‘…)(β—‘πΉβ€˜π‘¦)) ∧ (β—‘πΉβ€˜(π‘₯(2nd β€˜π‘†)𝑦)) = ((β—‘πΉβ€˜π‘₯)(2nd β€˜π‘…)(β—‘πΉβ€˜π‘¦))))))
1014, 18, 97, 100mpbir3and 1343 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) β†’ ◑𝐹 ∈ (𝑆 RngHom 𝑅))
1021ad2antll 728 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) β†’ ◑𝐹:ran (1st β€˜π‘†)–1-1-ontoβ†’ran (1st β€˜π‘…))
103101, 102jca 513 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) β†’ (◑𝐹 ∈ (𝑆 RngHom 𝑅) ∧ ◑𝐹:ran (1st β€˜π‘†)–1-1-ontoβ†’ran (1st β€˜π‘…)))
104103ex 414 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) β†’ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) β†’ (◑𝐹 ∈ (𝑆 RngHom 𝑅) ∧ ◑𝐹:ran (1st β€˜π‘†)–1-1-ontoβ†’ran (1st β€˜π‘…))))
10529, 12, 30, 37isrngoiso 36440 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) β†’ (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))))
10630, 37, 29, 12isrngoiso 36440 . . . 4 ((𝑆 ∈ RingOps ∧ 𝑅 ∈ RingOps) β†’ (◑𝐹 ∈ (𝑆 RngIso 𝑅) ↔ (◑𝐹 ∈ (𝑆 RngHom 𝑅) ∧ ◑𝐹:ran (1st β€˜π‘†)–1-1-ontoβ†’ran (1st β€˜π‘…))))
107106ancoms 460 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) β†’ (◑𝐹 ∈ (𝑆 RngIso 𝑅) ↔ (◑𝐹 ∈ (𝑆 RngHom 𝑅) ∧ ◑𝐹:ran (1st β€˜π‘†)–1-1-ontoβ†’ran (1st β€˜π‘…))))
108104, 105, 1073imtr4d 294 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) β†’ (𝐹 ∈ (𝑅 RngIso 𝑆) β†’ ◑𝐹 ∈ (𝑆 RngIso 𝑅)))
1091083impia 1118 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) β†’ ◑𝐹 ∈ (𝑆 RngIso 𝑅))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  β—‘ccnv 5633  ran crn 5635  βŸΆwf 6493  β€“1-1β†’wf1 6494  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  GIdcgi 29435  RingOpscrngo 36356   RngHom crnghom 36422   RngIso crngiso 36423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8768  df-grpo 29438  df-gid 29439  df-ablo 29490  df-ass 36305  df-exid 36307  df-mgmOLD 36311  df-sgrOLD 36323  df-mndo 36329  df-rngo 36357  df-rngohom 36425  df-rngoiso 36438
This theorem is referenced by:  riscer  36450
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