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| Mirrors > Home > ILE Home > Th. List > ghmnsgpreima | Unicode version | ||
| Description: The inverse image of a normal subgroup under a homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.) |
| Ref | Expression |
|---|---|
| ghmnsgpreima |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nsgsubg 13541 |
. . 3
| |
| 2 | ghmpreima 13602 |
. . 3
| |
| 3 | 1, 2 | sylan2 286 |
. 2
|
| 4 | ghmgrp1 13581 |
. . . . . 6
| |
| 5 | 4 | ad2antrr 488 |
. . . . 5
|
| 6 | simprl 529 |
. . . . . 6
| |
| 7 | simprr 531 |
. . . . . . . 8
| |
| 8 | simpll 527 |
. . . . . . . . . . 11
| |
| 9 | eqid 2205 |
. . . . . . . . . . . 12
| |
| 10 | eqid 2205 |
. . . . . . . . . . . 12
| |
| 11 | 9, 10 | ghmf 13583 |
. . . . . . . . . . 11
|
| 12 | 8, 11 | syl 14 |
. . . . . . . . . 10
|
| 13 | 12 | ffnd 5426 |
. . . . . . . . 9
|
| 14 | elpreima 5699 |
. . . . . . . . 9
| |
| 15 | 13, 14 | syl 14 |
. . . . . . . 8
|
| 16 | 7, 15 | mpbid 147 |
. . . . . . 7
|
| 17 | 16 | simpld 112 |
. . . . . 6
|
| 18 | eqid 2205 |
. . . . . . 7
| |
| 19 | 9, 18 | grpcl 13340 |
. . . . . 6
|
| 20 | 5, 6, 17, 19 | syl3anc 1250 |
. . . . 5
|
| 21 | eqid 2205 |
. . . . . 6
| |
| 22 | 9, 21 | grpsubcl 13412 |
. . . . 5
|
| 23 | 5, 20, 6, 22 | syl3anc 1250 |
. . . 4
|
| 24 | eqid 2205 |
. . . . . . . 8
| |
| 25 | 9, 21, 24 | ghmsub 13587 |
. . . . . . 7
|
| 26 | 8, 20, 6, 25 | syl3anc 1250 |
. . . . . 6
|
| 27 | eqid 2205 |
. . . . . . . . 9
| |
| 28 | 9, 18, 27 | ghmlin 13584 |
. . . . . . . 8
|
| 29 | 8, 6, 17, 28 | syl3anc 1250 |
. . . . . . 7
|
| 30 | 29 | oveq1d 5959 |
. . . . . 6
|
| 31 | 26, 30 | eqtrd 2238 |
. . . . 5
|
| 32 | simplr 528 |
. . . . . 6
| |
| 33 | 12, 6 | ffvelcdmd 5716 |
. . . . . 6
|
| 34 | 16 | simprd 114 |
. . . . . 6
|
| 35 | 10, 27, 24 | nsgconj 13542 |
. . . . . 6
|
| 36 | 32, 33, 34, 35 | syl3anc 1250 |
. . . . 5
|
| 37 | 31, 36 | eqeltrd 2282 |
. . . 4
|
| 38 | elpreima 5699 |
. . . . 5
| |
| 39 | 13, 38 | syl 14 |
. . . 4
|
| 40 | 23, 37, 39 | mpbir2and 947 |
. . 3
|
| 41 | 40 | ralrimivva 2588 |
. 2
|
| 42 | 9, 18, 21 | isnsg3 13543 |
. 2
|
| 43 | 3, 41, 42 | sylanbrc 417 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-addcom 8025 ax-addass 8027 ax-i2m1 8030 ax-0lt1 8031 ax-0id 8033 ax-rnegex 8034 ax-pre-ltirr 8037 ax-pre-ltadd 8041 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-pnf 8109 df-mnf 8110 df-ltxr 8112 df-inn 9037 df-2 9095 df-ndx 12835 df-slot 12836 df-base 12838 df-sets 12839 df-iress 12840 df-plusg 12922 df-0g 13090 df-mgm 13188 df-sgrp 13234 df-mnd 13249 df-grp 13335 df-minusg 13336 df-sbg 13337 df-subg 13506 df-nsg 13507 df-ghm 13577 |
| This theorem is referenced by: ghmker 13606 |
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