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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngoisohom | Structured version Visualization version GIF version |
Description: A ring isomorphism is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.) |
Ref | Expression |
---|---|
rngoisohom | β’ ((π β RingOps β§ π β RingOps β§ πΉ β (π RingOpsIso π)) β πΉ β (π RingOpsHom π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . . 4 β’ (1st βπ ) = (1st βπ ) | |
2 | eqid 2725 | . . . 4 β’ ran (1st βπ ) = ran (1st βπ ) | |
3 | eqid 2725 | . . . 4 β’ (1st βπ) = (1st βπ) | |
4 | eqid 2725 | . . . 4 β’ ran (1st βπ) = ran (1st βπ) | |
5 | 1, 2, 3, 4 | isrngoiso 37508 | . . 3 β’ ((π β RingOps β§ π β RingOps) β (πΉ β (π RingOpsIso π) β (πΉ β (π RingOpsHom π) β§ πΉ:ran (1st βπ )β1-1-ontoβran (1st βπ)))) |
6 | 5 | simprbda 497 | . 2 β’ (((π β RingOps β§ π β RingOps) β§ πΉ β (π RingOpsIso π)) β πΉ β (π RingOpsHom π)) |
7 | 6 | 3impa 1107 | 1 β’ ((π β RingOps β§ π β RingOps β§ πΉ β (π RingOpsIso π)) β πΉ β (π RingOpsHom π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 β wcel 2098 ran crn 5673 β1-1-ontoβwf1o 6542 βcfv 6543 (class class class)co 7416 1st c1st 7989 RingOpscrngo 37424 RingOpsHom crngohom 37490 RingOpsIso crngoiso 37491 |
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-pr 5423 |
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-rab 3420 df-v 3465 df-sbc 3769 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-rngoiso 37506 |
This theorem is referenced by: rngoisoco 37512 |
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