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Mathbox for Jeff Madsen |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rngogrphom | Structured version Visualization version GIF version | ||
| Description: A ring homomorphism is a group homomorphism. (Contributed by Jeff Madsen, 2-Jan-2011.) |
| Ref | Expression |
|---|---|
| rnggrphom.1 | β’ πΊ = (1st βπ ) |
| rnggrphom.2 | β’ π½ = (1st βπ) |
| Ref | Expression |
|---|---|
| rngogrphom | β’ ((π β RingOps β§ π β RingOps β§ πΉ β (π RingOpsHom π)) β πΉ β (πΊ GrpOpHom π½)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rnggrphom.1 | . . 3 β’ πΊ = (1st βπ ) | |
| 2 | eqid 2725 | . . 3 β’ ran πΊ = ran πΊ | |
| 3 | rnggrphom.2 | . . 3 β’ π½ = (1st βπ) | |
| 4 | eqid 2725 | . . 3 β’ ran π½ = ran π½ | |
| 5 | 1, 2, 3, 4 | rngohomf 37496 | . 2 β’ ((π β RingOps β§ π β RingOps β§ πΉ β (π RingOpsHom π)) β πΉ:ran πΊβΆran π½) |
| 6 | 1, 2, 3 | rngohomadd 37499 | . . . 4 β’ (((π β RingOps β§ π β RingOps β§ πΉ β (π RingOpsHom π)) β§ (π₯ β ran πΊ β§ π¦ β ran πΊ)) β (πΉβ(π₯πΊπ¦)) = ((πΉβπ₯)π½(πΉβπ¦))) |
| 7 | 6 | eqcomd 2731 | . . 3 β’ (((π β RingOps β§ π β RingOps β§ πΉ β (π RingOpsHom π)) β§ (π₯ β ran πΊ β§ π¦ β ran πΊ)) β ((πΉβπ₯)π½(πΉβπ¦)) = (πΉβ(π₯πΊπ¦))) |
| 8 | 7 | ralrimivva 3191 | . 2 β’ ((π β RingOps β§ π β RingOps β§ πΉ β (π RingOpsHom π)) β βπ₯ β ran πΊβπ¦ β ran πΊ((πΉβπ₯)π½(πΉβπ¦)) = (πΉβ(π₯πΊπ¦))) |
| 9 | 1 | rngogrpo 37440 | . . . 4 β’ (π β RingOps β πΊ β GrpOp) |
| 10 | 3 | rngogrpo 37440 | . . . 4 β’ (π β RingOps β π½ β GrpOp) |
| 11 | 2, 4 | elghomOLD 37417 | . . . 4 β’ ((πΊ β GrpOp β§ π½ β GrpOp) β (πΉ β (πΊ GrpOpHom π½) β (πΉ:ran πΊβΆran π½ β§ βπ₯ β ran πΊβπ¦ β ran πΊ((πΉβπ₯)π½(πΉβπ¦)) = (πΉβ(π₯πΊπ¦))))) |
| 12 | 9, 10, 11 | syl2an 594 | . . 3 β’ ((π β RingOps β§ π β RingOps) β (πΉ β (πΊ GrpOpHom π½) β (πΉ:ran πΊβΆran π½ β§ βπ₯ β ran πΊβπ¦ β ran πΊ((πΉβπ₯)π½(πΉβπ¦)) = (πΉβ(π₯πΊπ¦))))) |
| 13 | 12 | 3adant3 1129 | . 2 β’ ((π β RingOps β§ π β RingOps β§ πΉ β (π RingOpsHom π)) β (πΉ β (πΊ GrpOpHom π½) β (πΉ:ran πΊβΆran π½ β§ βπ₯ β ran πΊβπ¦ β ran πΊ((πΉβπ₯)π½(πΉβπ¦)) = (πΉβ(π₯πΊπ¦))))) |
| 14 | 5, 8, 13 | mpbir2and 711 | 1 β’ ((π β RingOps β§ π β RingOps β§ πΉ β (π RingOpsHom π)) β πΉ β (πΊ GrpOpHom π½)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3051 ran crn 5673 βΆwf 6539 βcfv 6543 (class class class)co 7416 1st c1st 7989 GrpOpcgr 30343 GrpOpHom cghomOLD 37413 RingOpscrngo 37424 RingOpsHom crngohom 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-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
| 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-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 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 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-1st 7991 df-2nd 7992 df-map 8845 df-ablo 30399 df-ghomOLD 37414 df-rngo 37425 df-rngohom 37493 |
| This theorem is referenced by: rngohom0 37502 rngohomsub 37503 rngokerinj 37505 |
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