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Theorem rngoisocnv 37510
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 RingOpsIso 𝑆)) β†’ ◑𝐹 ∈ (𝑆 RingOpsIso 𝑅))

Proof of Theorem rngoisocnv
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1ocnv 6845 . . . . . . . 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 6833 . . . . . . . 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 727 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 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 2725 . . . . . . . . . 10 (2nd β€˜π‘…) = (2nd β€˜π‘…)
6 eqid 2725 . . . . . . . . . 10 (GIdβ€˜(2nd β€˜π‘…)) = (GIdβ€˜(2nd β€˜π‘…))
7 eqid 2725 . . . . . . . . . 10 (2nd β€˜π‘†) = (2nd β€˜π‘†)
8 eqid 2725 . . . . . . . . . 10 (GIdβ€˜(2nd β€˜π‘†)) = (GIdβ€˜(2nd β€˜π‘†))
95, 6, 7, 8rngohom1 37497 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) β†’ (πΉβ€˜(GIdβ€˜(2nd β€˜π‘…))) = (GIdβ€˜(2nd β€˜π‘†)))
1093expa 1115 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) β†’ (πΉβ€˜(GIdβ€˜(2nd β€˜π‘…))) = (GIdβ€˜(2nd β€˜π‘†)))
1110adantrr 715 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 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 2725 . . . . . . . . . . 11 ran (1st β€˜π‘…) = ran (1st β€˜π‘…)
1312, 5, 6rngo1cl 37468 . . . . . . . . . 10 (𝑅 ∈ RingOps β†’ (GIdβ€˜(2nd β€˜π‘…)) ∈ ran (1st β€˜π‘…))
14 f1ocnvfv 7282 . . . . . . . . . 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 591 . . . . . . . . 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 457 . . . . . . . 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 747 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 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 RingOpsHom 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 7281 . . . . . . . . . . . . . 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 7281 . . . . . . . . . . . . . 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 617 . . . . . . . . . . . . 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 7424 . . . . . . . . . . . . 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 712 . . . . . . . . . . 11 (((𝐹 ∈ (𝑅 RingOpsHom 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 712 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 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 7289 . . . . . . . . . . . . . . . 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 7289 . . . . . . . . . . . . . . . 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 617 . . . . . . . . . . . . . . 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 2725 . . . . . . . . . . . . . . . 16 (1st β€˜π‘…) = (1st β€˜π‘…)
30 eqid 2725 . . . . . . . . . . . . . . . 16 (1st β€˜π‘†) = (1st β€˜π‘†)
3129, 12, 30rngohomadd 37498 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 591 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 419 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 1115 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 453 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 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 405 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 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 2725 . . . . . . . . . . . . . . . 16 ran (1st β€˜π‘†) = ran (1st β€˜π‘†)
3830, 37rngogcl 37441 . . . . . . . . . . . . . . 15 ((𝑆 ∈ RingOps ∧ π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†)) β†’ (π‘₯(1st β€˜π‘†)𝑦) ∈ ran (1st β€˜π‘†))
39383expb 1117 . . . . . . . . . . . . . 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 7281 . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . 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 578 . . . . . . . . . . . . 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 650 . . . . . . . . . . . 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 716 . . . . . . . . . . 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 718 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 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 2776 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 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 6832 . . . . . . . . . . . 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 725 . . . . . . . . . . 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 7289 . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . 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 578 . . . . . . . . . . . . 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 650 . . . . . . . . . . . 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 716 . . . . . . . . . . 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 37441 . . . . . . . . . . . . . . 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 1117 . . . . . . . . . . . . . 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 591 . . . . . . . . . . . . 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 466 . . . . . . . . . . . 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 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‘…)(β—‘πΉβ€˜π‘¦)) ∈ ran (1st β€˜π‘…))
59 f1fveq 7267 . . . . . . . . . . 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 835 . . . . . . . . . 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 718 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 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 RingOpsHom 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 7424 . . . . . . . . . . . . 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 712 . . . . . . . . . . 11 (((𝐹 ∈ (𝑅 RingOpsHom 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 712 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 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 37499 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 591 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 419 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 1115 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 453 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 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 405 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 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 37430 . . . . . . . . . . . . . . 15 ((𝑆 ∈ RingOps ∧ π‘₯ ∈ ran (1st β€˜π‘†) ∧ 𝑦 ∈ ran (1st β€˜π‘†)) β†’ (π‘₯(2nd β€˜π‘†)𝑦) ∈ ran (1st β€˜π‘†))
74733expb 1117 . . . . . . . . . . . . . 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 7281 . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . 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 578 . . . . . . . . . . . . 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 650 . . . . . . . . . . . 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 716 . . . . . . . . . . 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 718 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 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 2776 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 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 7289 . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . 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 578 . . . . . . . . . . . . 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 650 . . . . . . . . . . . 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 716 . . . . . . . . . . 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 37430 . . . . . . . . . . . . . . 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 1117 . . . . . . . . . . . . . 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 591 . . . . . . . . . . . . 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 466 . . . . . . . . . . . 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 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‘…)(β—‘πΉβ€˜π‘¦)) ∈ ran (1st β€˜π‘…))
92 f1fveq 7267 . . . . . . . . . . 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 835 . . . . . . . . . 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 718 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 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 RingOpsHom 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 510 . . . . . . 7 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 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 3191 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 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 37494 . . . . . . . 8 ((𝑆 ∈ RingOps ∧ 𝑅 ∈ RingOps) β†’ (◑𝐹 ∈ (𝑆 RingOpsHom 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 457 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) β†’ (◑𝐹 ∈ (𝑆 RingOpsHom 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 479 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) β†’ (◑𝐹 ∈ (𝑆 RingOpsHom 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 1339 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) β†’ ◑𝐹 ∈ (𝑆 RingOpsHom 𝑅))
1021ad2antll 727 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 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 510 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))) β†’ (◑𝐹 ∈ (𝑆 RingOpsHom 𝑅) ∧ ◑𝐹:ran (1st β€˜π‘†)–1-1-ontoβ†’ran (1st β€˜π‘…)))
104103ex 411 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) β†’ ((𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†)) β†’ (◑𝐹 ∈ (𝑆 RingOpsHom 𝑅) ∧ ◑𝐹:ran (1st β€˜π‘†)–1-1-ontoβ†’ran (1st β€˜π‘…))))
10529, 12, 30, 37isrngoiso 37507 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) β†’ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingOpsHom 𝑆) ∧ 𝐹:ran (1st β€˜π‘…)–1-1-ontoβ†’ran (1st β€˜π‘†))))
10630, 37, 29, 12isrngoiso 37507 . . . 4 ((𝑆 ∈ RingOps ∧ 𝑅 ∈ RingOps) β†’ (◑𝐹 ∈ (𝑆 RingOpsIso 𝑅) ↔ (◑𝐹 ∈ (𝑆 RingOpsHom 𝑅) ∧ ◑𝐹:ran (1st β€˜π‘†)–1-1-ontoβ†’ran (1st β€˜π‘…))))
107106ancoms 457 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) β†’ (◑𝐹 ∈ (𝑆 RingOpsIso 𝑅) ↔ (◑𝐹 ∈ (𝑆 RingOpsHom 𝑅) ∧ ◑𝐹:ran (1st β€˜π‘†)–1-1-ontoβ†’ran (1st β€˜π‘…))))
108104, 105, 1073imtr4d 293 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) β†’ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) β†’ ◑𝐹 ∈ (𝑆 RingOpsIso 𝑅)))
1091083impia 1114 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) β†’ ◑𝐹 ∈ (𝑆 RingOpsIso 𝑅))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051  β—‘ccnv 5671  ran crn 5673  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7415  1st c1st 7987  2nd c2nd 7988  GIdcgi 30342  RingOpscrngo 37423   RingOpsHom crngohom 37489   RingOpsIso crngoiso 37490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7989  df-2nd 7990  df-map 8843  df-grpo 30345  df-gid 30346  df-ablo 30397  df-ass 37372  df-exid 37374  df-mgmOLD 37378  df-sgrOLD 37390  df-mndo 37396  df-rngo 37424  df-rngohom 37492  df-rngoiso 37505
This theorem is referenced by:  riscer  37517
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