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Mirrors > Home > NFE Home > Th. List > caovmo | Unicode version |
Description: Uniqueness of inverse element in commutative, associative operation with identity. Remark in proof of Proposition 9-2.4 of [Gleason] p. 119. (Contributed by set.mm contributors, 4-Mar-1996.) |
Ref | Expression |
---|---|
caovmo.1 | |
caovmo.2 | |
caovmo.dom | |
caovmo.3 | |
caovmo.com | |
caovmo.ass | |
caovmo.id |
Ref | Expression |
---|---|
caovmo |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2413 | . . . . 5 | |
2 | oveq2 5531 | . . . . . 6 | |
3 | 2 | eqeq1d 2361 | . . . . 5 |
4 | 1, 3 | anbi12d 691 | . . . 4 |
5 | 4 | mo4 2237 | . . 3 |
6 | caovmo.1 | . . . . . . . . 9 | |
7 | vex 2862 | . . . . . . . . 9 | |
8 | vex 2862 | . . . . . . . . 9 | |
9 | caovmo.ass | . . . . . . . . 9 | |
10 | 6, 7, 8, 9 | caovass 5627 | . . . . . . . 8 |
11 | caovmo.com | . . . . . . . . 9 | |
12 | 6, 7, 8, 11, 9 | caov12 5636 | . . . . . . . 8 |
13 | 10, 12 | eqtri 2373 | . . . . . . 7 |
14 | oveq2 5531 | . . . . . . . 8 | |
15 | oveq1 5530 | . . . . . . . . . 10 | |
16 | id 19 | . . . . . . . . . 10 | |
17 | 15, 16 | eqeq12d 2367 | . . . . . . . . 9 |
18 | caovmo.id | . . . . . . . . 9 | |
19 | 17, 18 | vtoclga 2920 | . . . . . . . 8 |
20 | 14, 19 | sylan9eqr 2407 | . . . . . . 7 |
21 | 13, 20 | syl5eq 2397 | . . . . . 6 |
22 | 21 | ad2ant2rl 729 | . . . . 5 |
23 | oveq1 5530 | . . . . . . 7 | |
24 | caovmo.2 | . . . . . . . . . 10 | |
25 | 24 | elexi 2868 | . . . . . . . . 9 |
26 | 25, 8, 11 | caovcom 5625 | . . . . . . . 8 |
27 | oveq1 5530 | . . . . . . . . . 10 | |
28 | id 19 | . . . . . . . . . 10 | |
29 | 27, 28 | eqeq12d 2367 | . . . . . . . . 9 |
30 | 29, 18 | vtoclga 2920 | . . . . . . . 8 |
31 | 26, 30 | syl5eq 2397 | . . . . . . 7 |
32 | 23, 31 | sylan9eq 2405 | . . . . . 6 |
33 | 32 | ad2ant2lr 728 | . . . . 5 |
34 | 22, 33 | eqtr3d 2387 | . . . 4 |
35 | 34 | ax-gen 1546 | . . 3 |
36 | 5, 35 | mpgbir 1550 | . 2 |
37 | eleq1 2413 | . . . . . 6 | |
38 | 24, 37 | mpbiri 224 | . . . . 5 |
39 | caovmo.dom | . . . . . . 7 | |
40 | caovmo.3 | . . . . . . 7 | |
41 | 7, 39, 40 | ndmovrcl 5616 | . . . . . 6 |
42 | 41 | simprd 449 | . . . . 5 |
43 | 38, 42 | syl 15 | . . . 4 |
44 | 43 | ancri 535 | . . 3 |
45 | 44 | moimi 2251 | . 2 |
46 | 36, 45 | ax-mp 8 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 358 wal 1540 wceq 1642 wcel 1710 wmo 2205 cvv 2859 c0 3550 cxp 4770 cdm 4772 (class class class)co 5525 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-ima 4727 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-fv 4795 df-ov 5526 |
This theorem is referenced by: (None) |
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