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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngohom1 | Structured version Visualization version GIF version |
Description: A ring homomorphism preserves 1. (Contributed by Jeff Madsen, 24-Jun-2011.) |
Ref | Expression |
---|---|
rnghom1.1 | β’ π» = (2nd βπ ) |
rnghom1.2 | β’ π = (GIdβπ») |
rnghom1.3 | β’ πΎ = (2nd βπ) |
rnghom1.4 | β’ π = (GIdβπΎ) |
Ref | Expression |
---|---|
rngohom1 | β’ ((π β RingOps β§ π β RingOps β§ πΉ β (π RingOpsHom π)) β (πΉβπ) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . . 5 β’ (1st βπ ) = (1st βπ ) | |
2 | rnghom1.1 | . . . . 5 β’ π» = (2nd βπ ) | |
3 | eqid 2728 | . . . . 5 β’ ran (1st βπ ) = ran (1st βπ ) | |
4 | rnghom1.2 | . . . . 5 β’ π = (GIdβπ») | |
5 | eqid 2728 | . . . . 5 β’ (1st βπ) = (1st βπ) | |
6 | rnghom1.3 | . . . . 5 β’ πΎ = (2nd βπ) | |
7 | eqid 2728 | . . . . 5 β’ ran (1st βπ) = ran (1st βπ) | |
8 | rnghom1.4 | . . . . 5 β’ π = (GIdβπΎ) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | isrngohom 37471 | . . . 4 β’ ((π β RingOps β§ π β RingOps) β (πΉ β (π RingOpsHom π) β (πΉ:ran (1st βπ )βΆran (1st βπ) β§ (πΉβπ) = π β§ βπ₯ β ran (1st βπ )βπ¦ β ran (1st βπ )((πΉβ(π₯(1st βπ )π¦)) = ((πΉβπ₯)(1st βπ)(πΉβπ¦)) β§ (πΉβ(π₯π»π¦)) = ((πΉβπ₯)πΎ(πΉβπ¦)))))) |
10 | 9 | biimpa 475 | . . 3 β’ (((π β RingOps β§ π β RingOps) β§ πΉ β (π RingOpsHom π)) β (πΉ:ran (1st βπ )βΆran (1st βπ) β§ (πΉβπ) = π β§ βπ₯ β ran (1st βπ )βπ¦ β ran (1st βπ )((πΉβ(π₯(1st βπ )π¦)) = ((πΉβπ₯)(1st βπ)(πΉβπ¦)) β§ (πΉβ(π₯π»π¦)) = ((πΉβπ₯)πΎ(πΉβπ¦))))) |
11 | 10 | simp2d 1140 | . 2 β’ (((π β RingOps β§ π β RingOps) β§ πΉ β (π RingOpsHom π)) β (πΉβπ) = π) |
12 | 11 | 3impa 1107 | 1 β’ ((π β RingOps β§ π β RingOps β§ πΉ β (π RingOpsHom π)) β (πΉβπ) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3058 ran crn 5683 βΆwf 6549 βcfv 6553 (class class class)co 7426 1st c1st 7997 2nd c2nd 7998 GIdcgi 30320 RingOpscrngo 37400 RingOpsHom crngohom 37466 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-map 8853 df-rngohom 37469 |
This theorem is referenced by: rngohomco 37480 rngoisocnv 37487 |
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