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Mirrors > Home > ILE Home > Th. List > grprinvlem | Unicode version |
Description: Lemma for grprinvd 5716. (Contributed by NM, 9-Aug-2013.) |
Ref | Expression |
---|---|
grprinvlem.c | |
grprinvlem.o | |
grprinvlem.i | |
grprinvlem.a | |
grprinvlem.n | |
grprinvlem.x | |
grprinvlem.e |
Ref | Expression |
---|---|
grprinvlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grprinvlem.x | . . 3 | |
2 | grprinvlem.n | . . . . . 6 | |
3 | 2 | ralrimiva 2434 | . . . . 5 |
4 | oveq2 5540 | . . . . . . . 8 | |
5 | 4 | eqeq1d 2089 | . . . . . . 7 |
6 | 5 | rexbidv 2369 | . . . . . 6 |
7 | 6 | cbvralv 2577 | . . . . 5 |
8 | 3, 7 | sylib 120 | . . . 4 |
9 | oveq2 5540 | . . . . . . 7 | |
10 | 9 | eqeq1d 2089 | . . . . . 6 |
11 | 10 | rexbidv 2369 | . . . . 5 |
12 | 11 | rspccva 2700 | . . . 4 |
13 | 8, 12 | sylan 277 | . . 3 |
14 | 1, 13 | syldan 276 | . 2 |
15 | grprinvlem.e | . . . . 5 | |
16 | 15 | oveq2d 5548 | . . . 4 |
17 | 16 | adantr 270 | . . 3 |
18 | simprr 498 | . . . . 5 | |
19 | 18 | oveq1d 5547 | . . . 4 |
20 | simpll 495 | . . . . . 6 | |
21 | grprinvlem.a | . . . . . . 7 | |
22 | 21 | caovassg 5679 | . . . . . 6 |
23 | 20, 22 | sylan 277 | . . . . 5 |
24 | simprl 497 | . . . . 5 | |
25 | 1 | adantr 270 | . . . . 5 |
26 | 23, 24, 25, 25 | caovassd 5680 | . . . 4 |
27 | grprinvlem.i | . . . . . . . . 9 | |
28 | 27 | ralrimiva 2434 | . . . . . . . 8 |
29 | oveq2 5540 | . . . . . . . . . 10 | |
30 | id 19 | . . . . . . . . . 10 | |
31 | 29, 30 | eqeq12d 2095 | . . . . . . . . 9 |
32 | 31 | cbvralv 2577 | . . . . . . . 8 |
33 | 28, 32 | sylib 120 | . . . . . . 7 |
34 | 33 | adantr 270 | . . . . . 6 |
35 | oveq2 5540 | . . . . . . . 8 | |
36 | id 19 | . . . . . . . 8 | |
37 | 35, 36 | eqeq12d 2095 | . . . . . . 7 |
38 | 37 | rspcv 2697 | . . . . . 6 |
39 | 1, 34, 38 | sylc 61 | . . . . 5 |
40 | 39 | adantr 270 | . . . 4 |
41 | 19, 26, 40 | 3eqtr3d 2121 | . . 3 |
42 | 17, 41, 18 | 3eqtr3d 2121 | . 2 |
43 | 14, 42 | rexlimddv 2481 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 w3a 919 wceq 1284 wcel 1433 wral 2348 wrex 2349 (class class class)co 5532 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 |
This theorem is referenced by: grprinvd 5716 |
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