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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 RngHom π)) β (πΉβπ) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . . 5 β’ (1st βπ ) = (1st βπ ) | |
2 | rnghom1.1 | . . . . 5 β’ π» = (2nd βπ ) | |
3 | eqid 2737 | . . . . 5 β’ ran (1st βπ ) = ran (1st βπ ) | |
4 | rnghom1.2 | . . . . 5 β’ π = (GIdβπ») | |
5 | eqid 2737 | . . . . 5 β’ (1st βπ) = (1st βπ) | |
6 | rnghom1.3 | . . . . 5 β’ πΎ = (2nd βπ) | |
7 | eqid 2737 | . . . . 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 36427 | . . . 4 β’ ((π β RingOps β§ π β RingOps) β (πΉ β (π RngHom 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 478 | . . 3 β’ (((π β RingOps β§ π β RingOps) β§ πΉ β (π RngHom 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 1144 | . 2 β’ (((π β RingOps β§ π β RingOps) β§ πΉ β (π RngHom π)) β (πΉβπ) = π) |
12 | 11 | 3impa 1111 | 1 β’ ((π β RingOps β§ π β RingOps β§ πΉ β (π RngHom π)) β (πΉβπ) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3065 ran crn 5635 βΆwf 6493 βcfv 6497 (class class class)co 7358 1st c1st 7920 2nd c2nd 7921 GIdcgi 29435 RingOpscrngo 36356 RngHom crnghom 36422 |
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-rab 3409 df-v 3448 df-sbc 3741 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-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-map 8768 df-rngohom 36425 |
This theorem is referenced by: rngohomco 36436 rngoisocnv 36443 |
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