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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 RngHom π)) β πΉ β (πΊ GrpOpHom π½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnggrphom.1 | . . 3 β’ πΊ = (1st βπ ) | |
2 | eqid 2737 | . . 3 β’ ran πΊ = ran πΊ | |
3 | rnggrphom.2 | . . 3 β’ π½ = (1st βπ) | |
4 | eqid 2737 | . . 3 β’ ran π½ = ran π½ | |
5 | 1, 2, 3, 4 | rngohomf 36428 | . 2 β’ ((π β RingOps β§ π β RingOps β§ πΉ β (π RngHom π)) β πΉ:ran πΊβΆran π½) |
6 | 1, 2, 3 | rngohomadd 36431 | . . . 4 β’ (((π β RingOps β§ π β RingOps β§ πΉ β (π RngHom π)) β§ (π₯ β ran πΊ β§ π¦ β ran πΊ)) β (πΉβ(π₯πΊπ¦)) = ((πΉβπ₯)π½(πΉβπ¦))) |
7 | 6 | eqcomd 2743 | . . 3 β’ (((π β RingOps β§ π β RingOps β§ πΉ β (π RngHom π)) β§ (π₯ β ran πΊ β§ π¦ β ran πΊ)) β ((πΉβπ₯)π½(πΉβπ¦)) = (πΉβ(π₯πΊπ¦))) |
8 | 7 | ralrimivva 3198 | . 2 β’ ((π β RingOps β§ π β RingOps β§ πΉ β (π RngHom π)) β βπ₯ β ran πΊβπ¦ β ran πΊ((πΉβπ₯)π½(πΉβπ¦)) = (πΉβ(π₯πΊπ¦))) |
9 | 1 | rngogrpo 36372 | . . . 4 β’ (π β RingOps β πΊ β GrpOp) |
10 | 3 | rngogrpo 36372 | . . . 4 β’ (π β RingOps β π½ β GrpOp) |
11 | 2, 4 | elghomOLD 36349 | . . . 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 597 | . . 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 1133 | . 2 β’ ((π β RingOps β§ π β RingOps β§ πΉ β (π RngHom 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 712 | 1 β’ ((π β RingOps β§ π β RingOps β§ πΉ β (π RngHom π)) β πΉ β (πΊ GrpOpHom π½)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 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 GrpOpcgr 29434 GrpOpHom cghomOLD 36345 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-rep 5243 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-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 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-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-map 8768 df-ablo 29490 df-ghomOLD 36346 df-rngo 36357 df-rngohom 36425 |
This theorem is referenced by: rngohom0 36434 rngohomsub 36435 rngokerinj 36437 |
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