![]() |
Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
|
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 2726 | . . 3 β’ ran πΊ = ran πΊ | |
3 | rnggrphom.2 | . . 3 β’ π½ = (1st βπ) | |
4 | eqid 2726 | . . 3 β’ ran π½ = ran π½ | |
5 | 1, 2, 3, 4 | rngohomf 37347 | . 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 37350 | . . . 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 2732 | . . 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 3194 | . 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 37291 | . . . 4 β’ (π β RingOps β πΊ β GrpOp) |
10 | 3 | rngogrpo 37291 | . . . 4 β’ (π β RingOps β π½ β GrpOp) |
11 | 2, 4 | elghomOLD 37268 | . . . 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 595 | . . 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 710 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3055 ran crn 5670 βΆwf 6533 βcfv 6537 (class class class)co 7405 1st c1st 7972 GrpOpcgr 30251 GrpOpHom cghomOLD 37264 RingOpscrngo 37275 RingOpsHom crngohom 37341 |
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-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
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-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 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-1st 7974 df-2nd 7975 df-map 8824 df-ablo 30307 df-ghomOLD 37265 df-rngo 37276 df-rngohom 37344 |
This theorem is referenced by: rngohom0 37353 rngohomsub 37354 rngokerinj 37356 |
Copyright terms: Public domain | W3C validator |