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Mathbox for Paul Chapman |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ghomsn | Structured version Unicode version |
Description: The endomorphism of the trivial group. (Contributed by Paul Chapman, 25-Feb-2008.) |
Ref | Expression |
---|---|
ghomsn.1 |
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ghomsn.2 |
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Ref | Expression |
---|---|
ghomsn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oi 5777 |
. . 3
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2 | f1of 5742 |
. . 3
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3 | 1, 2 | ax-mp 5 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | elsn 3992 |
. . . . 5
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5 | elsn 3992 |
. . . . 5
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6 | fveq2 5792 |
. . . . . . . 8
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7 | ghomsn.1 |
. . . . . . . . . 10
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8 | 7 | snid 4006 |
. . . . . . . . 9
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9 | fvresi 6006 |
. . . . . . . . 9
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10 | 8, 9 | ax-mp 5 |
. . . . . . . 8
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11 | 6, 10 | syl6eq 2508 |
. . . . . . 7
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12 | fveq2 5792 |
. . . . . . . 8
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13 | 12, 10 | syl6eq 2508 |
. . . . . . 7
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14 | 11, 13 | oveqan12d 6212 |
. . . . . 6
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15 | oveq12 6202 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 14, 15 | eqtr4d 2495 |
. . . . 5
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17 | 4, 5, 16 | syl2anb 479 |
. . . 4
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18 | ghomsn.2 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 7 | grposn 23847 |
. . . . . . 7
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20 | 18, 19 | eqeltri 2535 |
. . . . . 6
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21 | 18 | rneqi 5167 |
. . . . . . . 8
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22 | opex 4657 |
. . . . . . . . 9
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23 | 22 | rnsnop 5421 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 21, 23 | eqtr2i 2481 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 24 | grpocl 23832 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 20, 25 | mp3an1 1302 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | fvresi 6006 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 26, 27 | syl 16 |
. . . 4
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29 | 17, 28 | eqtr4d 2495 |
. . 3
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30 | 29 | rgen2a 2893 |
. 2
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31 | 24, 24 | elghom 23995 |
. . 3
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32 | 20, 20, 31 | mp2an 672 |
. 2
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33 | 3, 30, 32 | mpbir2an 911 |
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 1952 ax-ext 2430 ax-rep 4504 ax-sep 4514 ax-nul 4522 ax-pow 4571 ax-pr 4632 ax-un 6475 |
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 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-ral 2800 df-rex 2801 df-reu 2802 df-rab 2804 df-v 3073 df-sbc 3288 df-csb 3390 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-nul 3739 df-if 3893 df-pw 3963 df-sn 3979 df-pr 3981 df-op 3985 df-uni 4193 df-iun 4274 df-br 4394 df-opab 4452 df-mpt 4453 df-id 4737 df-xp 4947 df-rel 4948 df-cnv 4949 df-co 4950 df-dm 4951 df-rn 4952 df-res 4953 df-ima 4954 df-iota 5482 df-fun 5521 df-fn 5522 df-f 5523 df-f1 5524 df-fo 5525 df-f1o 5526 df-fv 5527 df-ov 6196 df-oprab 6197 df-mpt2 6198 df-grpo 23823 df-ghom 23990 |
This theorem is referenced by: ghomgrplem 27445 |
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