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Mirrors > Home > MPE Home > Th. List > ghmco | Structured version Unicode version |
Description: The composition of group homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) |
Ref | Expression |
---|---|
ghmco |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ghmmhm 15875 |
. . 3
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2 | ghmmhm 15875 |
. . 3
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3 | mhmco 15608 |
. . 3
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4 | 1, 2, 3 | syl2an 477 |
. 2
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5 | ghmgrp1 15867 |
. . 3
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6 | ghmgrp2 15868 |
. . 3
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7 | ghmmhmb 15876 |
. . 3
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8 | 5, 6, 7 | syl2anr 478 |
. 2
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9 | 4, 8 | eleqtrrd 2545 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4510 ax-sep 4520 ax-nul 4528 ax-pow 4577 ax-pr 4638 ax-un 6481 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2649 df-ral 2803 df-rex 2804 df-reu 2805 df-rmo 2806 df-rab 2807 df-v 3078 df-sbc 3293 df-csb 3395 df-dif 3438 df-un 3440 df-in 3442 df-ss 3449 df-nul 3745 df-if 3899 df-pw 3969 df-sn 3985 df-pr 3987 df-op 3991 df-uni 4199 df-iun 4280 df-br 4400 df-opab 4458 df-mpt 4459 df-id 4743 df-xp 4953 df-rel 4954 df-cnv 4955 df-co 4956 df-dm 4957 df-rn 4958 df-res 4959 df-ima 4960 df-iota 5488 df-fun 5527 df-fn 5528 df-f 5529 df-f1 5530 df-fo 5531 df-f1o 5532 df-fv 5533 df-riota 6160 df-ov 6202 df-oprab 6203 df-mpt2 6204 df-map 7325 df-0g 14498 df-mnd 15533 df-mhm 15582 df-grp 15663 df-ghm 15863 |
This theorem is referenced by: gimco 15914 rhmco 16947 lmhmco 17246 lmhmvsca 17248 frgpcyg 18130 nmoco 20447 nghmco 20448 |
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