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Mirrors > Home > MPE Home > Th. List > omopthlem1 | Structured version Visualization version Unicode version |
Description: Lemma for omopthi 7737. (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
omopthlem1.1 | |
omopthlem1.2 |
Ref | Expression |
---|---|
omopthlem1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omopthlem1.1 | . . . . 5 | |
2 | peano2 7086 | . . . . 5 | |
3 | 1, 2 | ax-mp 5 | . . . 4 |
4 | omopthlem1.2 | . . . 4 | |
5 | nnmwordi 7715 | . . . 4 | |
6 | 3, 4, 3, 5 | mp3an 1424 | . . 3 |
7 | nnmwordri 7716 | . . . 4 | |
8 | 3, 4, 4, 7 | mp3an 1424 | . . 3 |
9 | 6, 8 | sstrd 3613 | . 2 |
10 | 1 | nnoni 7072 | . . 3 |
11 | 4 | nnoni 7072 | . . 3 |
12 | 10, 11 | onsucssi 7041 | . 2 |
13 | 1, 1 | nnmcli 7695 | . . . . . 6 |
14 | 2onn 7720 | . . . . . . 7 | |
15 | 1, 14 | nnmcli 7695 | . . . . . 6 |
16 | 13, 15 | nnacli 7694 | . . . . 5 |
17 | 16 | nnoni 7072 | . . . 4 |
18 | 4, 4 | nnmcli 7695 | . . . . 5 |
19 | 18 | nnoni 7072 | . . . 4 |
20 | 17, 19 | onsucssi 7041 | . . 3 |
21 | 3, 1 | nnmcli 7695 | . . . . . 6 |
22 | nnasuc 7686 | . . . . . 6 | |
23 | 21, 1, 22 | mp2an 708 | . . . . 5 |
24 | nnmsuc 7687 | . . . . . 6 | |
25 | 3, 1, 24 | mp2an 708 | . . . . 5 |
26 | nnaass 7702 | . . . . . . . 8 | |
27 | 13, 1, 1, 26 | mp3an 1424 | . . . . . . 7 |
28 | nnmcom 7706 | . . . . . . . . . 10 | |
29 | 3, 1, 28 | mp2an 708 | . . . . . . . . 9 |
30 | nnmsuc 7687 | . . . . . . . . . 10 | |
31 | 1, 1, 30 | mp2an 708 | . . . . . . . . 9 |
32 | 29, 31 | eqtri 2644 | . . . . . . . 8 |
33 | 32 | oveq1i 6660 | . . . . . . 7 |
34 | nnm2 7729 | . . . . . . . . 9 | |
35 | 1, 34 | ax-mp 5 | . . . . . . . 8 |
36 | 35 | oveq2i 6661 | . . . . . . 7 |
37 | 27, 33, 36 | 3eqtr4ri 2655 | . . . . . 6 |
38 | suceq 5790 | . . . . . 6 | |
39 | 37, 38 | ax-mp 5 | . . . . 5 |
40 | 23, 25, 39 | 3eqtr4ri 2655 | . . . 4 |
41 | 40 | sseq1i 3629 | . . 3 |
42 | 20, 41 | bitri 264 | . 2 |
43 | 9, 12, 42 | 3imtr4i 281 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 wss 3574 csuc 5725 (class class class)co 6650 com 7065 c2o 7554 coa 7557 comu 7558 |
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-8 1992 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-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 |
This theorem is referenced by: omopthlem2 7736 |
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