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Mirrors > Home > ILE Home > Th. List > fimax2gtrilemstep | Unicode version |
Description: Lemma for fimax2gtri 6858. The induction step. (Contributed by Jim Kingdon, 5-Sep-2022.) |
Ref | Expression |
---|---|
fimax2gtri.po | |
fimax2gtri.tri | |
fimax2gtri.fin | |
fimax2gtri.n0 | |
fimax2gtri.ufin | |
fimax2gtri.uss | |
fimax2gtri.za | |
fimax2gtri.va | |
fimax2gtri.vu | |
fimax2gtri.zb |
Ref | Expression |
---|---|
fimax2gtrilemstep |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fimax2gtri.va | . . 3 | |
2 | fimax2gtri.za | . . 3 | |
3 | fimax2gtri.po | . . . 4 | |
4 | fimax2gtri.tri | . . . 4 | |
5 | 3, 4, 2, 1 | tridc 6856 | . . 3 DECID |
6 | 1, 2, 5 | ifcldcd 3550 | . 2 |
7 | simplr 520 | . . . . . . . 8 | |
8 | simpr 109 | . . . . . . . . . . . 12 | |
9 | 8 | iftrued 3522 | . . . . . . . . . . 11 |
10 | 9 | breq1d 3986 | . . . . . . . . . 10 |
11 | 10 | biimpa 294 | . . . . . . . . 9 |
12 | 11 | adantllr 473 | . . . . . . . 8 |
13 | 3 | ad2antrr 480 | . . . . . . . . . 10 |
14 | 2 | ad2antrr 480 | . . . . . . . . . 10 |
15 | 1 | ad2antrr 480 | . . . . . . . . . 10 |
16 | fimax2gtri.uss | . . . . . . . . . . . 12 | |
17 | 16 | ad2antrr 480 | . . . . . . . . . . 11 |
18 | simplr 520 | . . . . . . . . . . 11 | |
19 | 17, 18 | sseldd 3138 | . . . . . . . . . 10 |
20 | potr 4280 | . . . . . . . . . 10 | |
21 | 13, 14, 15, 19, 20 | syl13anc 1229 | . . . . . . . . 9 |
22 | 21 | adantr 274 | . . . . . . . 8 |
23 | 7, 12, 22 | mp2and 430 | . . . . . . 7 |
24 | fimax2gtri.zb | . . . . . . . . . 10 | |
25 | breq2 3980 | . . . . . . . . . . . 12 | |
26 | 25 | notbid 657 | . . . . . . . . . . 11 |
27 | 26 | cbvralv 2689 | . . . . . . . . . 10 |
28 | 24, 27 | sylib 121 | . . . . . . . . 9 |
29 | 28 | r19.21bi 2552 | . . . . . . . 8 |
30 | 29 | ad2antrr 480 | . . . . . . 7 |
31 | 23, 30 | pm2.65da 651 | . . . . . 6 |
32 | 29 | adantr 274 | . . . . . . 7 |
33 | simpr 109 | . . . . . . . . . 10 | |
34 | 33 | iffalsed 3525 | . . . . . . . . 9 |
35 | 34 | breq1d 3986 | . . . . . . . 8 |
36 | 35 | adantlr 469 | . . . . . . 7 |
37 | 32, 36 | mtbird 663 | . . . . . 6 |
38 | exmiddc 826 | . . . . . . . 8 DECID | |
39 | 5, 38 | syl 14 | . . . . . . 7 |
40 | 39 | adantr 274 | . . . . . 6 |
41 | 31, 37, 40 | mpjaodan 788 | . . . . 5 |
42 | 41 | ralrimiva 2537 | . . . 4 |
43 | breq2 3980 | . . . . . 6 | |
44 | 43 | notbid 657 | . . . . 5 |
45 | 44 | cbvralv 2689 | . . . 4 |
46 | 42, 45 | sylib 121 | . . 3 |
47 | 3 | adantr 274 | . . . . . . 7 |
48 | 1 | adantr 274 | . . . . . . 7 |
49 | poirr 4279 | . . . . . . 7 | |
50 | 47, 48, 49 | syl2anc 409 | . . . . . 6 |
51 | 9 | breq1d 3986 | . . . . . 6 |
52 | 50, 51 | mtbird 663 | . . . . 5 |
53 | 34 | breq1d 3986 | . . . . . 6 |
54 | 33, 53 | mtbird 663 | . . . . 5 |
55 | 52, 54, 39 | mpjaodan 788 | . . . 4 |
56 | breq2 3980 | . . . . . . 7 | |
57 | 56 | notbid 657 | . . . . . 6 |
58 | 57 | ralsng 3610 | . . . . 5 |
59 | 1, 58 | syl 14 | . . . 4 |
60 | 55, 59 | mpbird 166 | . . 3 |
61 | ralun 3299 | . . 3 | |
62 | 46, 60, 61 | syl2anc 409 | . 2 |
63 | breq1 3979 | . . . . 5 | |
64 | 63 | notbid 657 | . . . 4 |
65 | 64 | ralbidv 2464 | . . 3 |
66 | 65 | rspcev 2825 | . 2 |
67 | 6, 62, 66 | syl2anc 409 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 DECID wdc 824 w3o 966 wceq 1342 wcel 2135 wne 2334 wral 2442 wrex 2443 cun 3109 wss 3111 c0 3404 cif 3515 csn 3570 class class class wbr 3976 wpo 4266 cfn 6697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-if 3516 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-po 4268 |
This theorem is referenced by: fimax2gtri 6858 |
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