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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 2726 | . . . 4 β’ (1st βπ ) = (1st βπ ) | |
2 | eqid 2726 | . . . 4 β’ ran (1st βπ ) = ran (1st βπ ) | |
3 | eqid 2726 | . . . 4 β’ (1st βπ) = (1st βπ) | |
4 | eqid 2726 | . . . 4 β’ ran (1st βπ) = ran (1st βπ) | |
5 | 1, 2, 3, 4 | isrngoiso 37359 | . . 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 498 | . 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 395 β§ w3a 1084 β wcel 2098 ran crn 5670 β1-1-ontoβwf1o 6536 βcfv 6537 (class class class)co 7405 1st c1st 7972 RingOpscrngo 37275 RingOpsHom crngohom 37341 RingOpsIso crngoiso 37342 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-rngoiso 37357 |
This theorem is referenced by: rngoisoco 37363 |
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