Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mgm2nsgrplem2 | Structured version Visualization version Unicode version |
Description: Lemma 2 for mgm2nsgrp 17409. (Contributed by AV, 27-Jan-2020.) |
Ref | Expression |
---|---|
mgm2nsgrp.s | |
mgm2nsgrp.b | |
mgm2nsgrp.o | |
mgm2nsgrp.p |
Ref | Expression |
---|---|
mgm2nsgrplem2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prid1g 4295 | . . 3 | |
2 | mgm2nsgrp.s | . . 3 | |
3 | 1, 2 | syl6eleqr 2712 | . 2 |
4 | prid2g 4296 | . . 3 | |
5 | 4, 2 | syl6eleqr 2712 | . 2 |
6 | mgm2nsgrp.p | . . . . 5 | |
7 | mgm2nsgrp.o | . . . . 5 | |
8 | 6, 7 | eqtri 2644 | . . . 4 |
9 | 8 | a1i 11 | . . 3 |
10 | ifeq1 4090 | . . . . . . 7 | |
11 | ifid 4125 | . . . . . . 7 | |
12 | 10, 11 | syl6eq 2672 | . . . . . 6 |
13 | 12 | a1d 25 | . . . . 5 |
14 | eqeq1 2626 | . . . . . . . . . . 11 | |
15 | 14 | bicomd 213 | . . . . . . . . . 10 |
16 | 15 | notbid 308 | . . . . . . . . 9 |
17 | 16 | biimpac 503 | . . . . . . . 8 |
18 | 17 | intnand 962 | . . . . . . 7 |
19 | 18 | iffalsed 4097 | . . . . . 6 |
20 | 19 | ex 450 | . . . . 5 |
21 | 13, 20 | pm2.61i 176 | . . . 4 |
22 | 21 | ad2antll 765 | . . 3 |
23 | iftrue 4092 | . . . . . 6 | |
24 | 23 | adantl 482 | . . . . 5 |
25 | simpl 473 | . . . . 5 | |
26 | simpr 477 | . . . . 5 | |
27 | 9, 24, 25, 25, 26 | ovmpt2d 6788 | . . . 4 |
28 | 27, 26 | eqeltrd 2701 | . . 3 |
29 | 9, 22, 28, 26, 25 | ovmpt2d 6788 | . 2 |
30 | 3, 5, 29 | syl2an 494 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 384 wceq 1483 wcel 1990 cif 4086 cpr 4179 cfv 5888 (class class class)co 6650 cmpt2 6652 cbs 15857 cplusg 15941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 |
This theorem is referenced by: mgm2nsgrplem4 17408 |
Copyright terms: Public domain | W3C validator |