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Mirrors > Home > NFE Home > Th. List > addccan2nclem2 | Unicode version |
Description: Lemma for addccan2nc 6265. Establish stratification for induction. (Contributed by Scott Fenton, 2-Aug-2019.) |
Ref | Expression |
---|---|
addccan2nclem2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unab 3521 | . . 3 | |
2 | complab 3524 | . . . 4 ∼ | |
3 | 2 | uneq1i 3414 | . . 3 ∼ |
4 | imor 401 | . . . 4 | |
5 | 4 | abbii 2465 | . . 3 |
6 | 1, 3, 5 | 3eqtr4i 2383 | . 2 ∼ |
7 | addceq2 4384 | . . . . . . . 8 | |
8 | 7 | eqeq1d 2361 | . . . . . . 7 |
9 | 8 | abbidv 2467 | . . . . . 6 |
10 | 9 | eleq1d 2419 | . . . . 5 |
11 | addceq2 4384 | . . . . . . . 8 | |
12 | 11 | eqeq2d 2364 | . . . . . . 7 |
13 | 12 | abbidv 2467 | . . . . . 6 |
14 | 13 | eleq1d 2419 | . . . . 5 |
15 | elfix 5787 | . . . . . . . 8 AddC AddC AddC AddC | |
16 | brco 4883 | . . . . . . . . 9 AddC AddC AddC AddC | |
17 | addccan2nclem1 6263 | . . . . . . . . . . 11 AddC | |
18 | brcnv 4892 | . . . . . . . . . . . 12 AddC AddC | |
19 | addccan2nclem1 6263 | . . . . . . . . . . . 12 AddC | |
20 | 18, 19 | bitri 240 | . . . . . . . . . . 11 AddC |
21 | 17, 20 | anbi12i 678 | . . . . . . . . . 10 AddC AddC |
22 | 21 | exbii 1582 | . . . . . . . . 9 AddC AddC |
23 | 16, 22 | bitri 240 | . . . . . . . 8 AddC AddC |
24 | vex 2862 | . . . . . . . . . 10 | |
25 | vex 2862 | . . . . . . . . . 10 | |
26 | 24, 25 | addcex 4394 | . . . . . . . . 9 |
27 | eqeq1 2359 | . . . . . . . . 9 | |
28 | 26, 27 | ceqsexv 2894 | . . . . . . . 8 |
29 | 15, 23, 28 | 3bitri 262 | . . . . . . 7 AddC AddC |
30 | 29 | abbi2i 2464 | . . . . . 6 AddC AddC |
31 | addcfnex 5824 | . . . . . . . . . 10 AddC | |
32 | 1stex 4739 | . . . . . . . . . . . 12 | |
33 | vvex 4109 | . . . . . . . . . . . . 13 | |
34 | snex 4111 | . . . . . . . . . . . . 13 | |
35 | 33, 34 | xpex 5115 | . . . . . . . . . . . 12 |
36 | 32, 35 | resex 5117 | . . . . . . . . . . 11 |
37 | 36 | cnvex 5102 | . . . . . . . . . 10 |
38 | 31, 37 | coex 4750 | . . . . . . . . 9 AddC |
39 | 38 | cnvex 5102 | . . . . . . . 8 AddC |
40 | snex 4111 | . . . . . . . . . . . 12 | |
41 | 33, 40 | xpex 5115 | . . . . . . . . . . 11 |
42 | 32, 41 | resex 5117 | . . . . . . . . . 10 |
43 | 42 | cnvex 5102 | . . . . . . . . 9 |
44 | 31, 43 | coex 4750 | . . . . . . . 8 AddC |
45 | 39, 44 | coex 4750 | . . . . . . 7 AddC AddC |
46 | 45 | fixex 5789 | . . . . . 6 AddC AddC |
47 | 30, 46 | eqeltrri 2424 | . . . . 5 |
48 | 10, 14, 47 | vtocl2g 2918 | . . . 4 |
49 | complexg 4099 | . . . 4 ∼ | |
50 | 48, 49 | syl 15 | . . 3 ∼ |
51 | abexv 4324 | . . 3 | |
52 | unexg 4101 | . . 3 ∼ ∼ | |
53 | 50, 51, 52 | sylancl 643 | . 2 ∼ |
54 | 6, 53 | syl5eqelr 2438 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wo 357 wa 358 wex 1541 wceq 1642 wcel 1710 cab 2339 cvv 2859 ∼ ccompl 3205 cun 3207 csn 3737 cplc 4375 class class class wbr 4639 c1st 4717 ccom 4721 cxp 4770 ccnv 4771 cres 4774 cfix 5739 AddC caddcfn 5745 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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-csb 3137 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-iun 3971 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-1st 4723 df-swap 4724 df-sset 4725 df-co 4726 df-ima 4727 df-si 4728 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-fo 4793 df-fv 4795 df-2nd 4797 df-ov 5526 df-oprab 5528 df-mpt 5652 df-mpt2 5654 df-txp 5736 df-fix 5740 df-cup 5742 df-disj 5744 df-addcfn 5746 df-ins2 5750 df-ins3 5752 df-ins4 5756 df-si3 5758 |
This theorem is referenced by: addccan2nc 6265 |
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