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Mirrors > Home > ILE Home > Th. List > fimax2gtrilemstep | Unicode version |
Description: Lemma for fimax2gtri 6795. 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 6793 | . . 3 DECID |
6 | 1, 2, 5 | ifcldcd 3507 | . 2 |
7 | simplr 519 | . . . . . . . 8 | |
8 | simpr 109 | . . . . . . . . . . . 12 | |
9 | 8 | iftrued 3481 | . . . . . . . . . . 11 |
10 | 9 | breq1d 3939 | . . . . . . . . . 10 |
11 | 10 | biimpa 294 | . . . . . . . . 9 |
12 | 11 | adantllr 472 | . . . . . . . 8 |
13 | 3 | ad2antrr 479 | . . . . . . . . . 10 |
14 | 2 | ad2antrr 479 | . . . . . . . . . 10 |
15 | 1 | ad2antrr 479 | . . . . . . . . . 10 |
16 | fimax2gtri.uss | . . . . . . . . . . . 12 | |
17 | 16 | ad2antrr 479 | . . . . . . . . . . 11 |
18 | simplr 519 | . . . . . . . . . . 11 | |
19 | 17, 18 | sseldd 3098 | . . . . . . . . . 10 |
20 | potr 4230 | . . . . . . . . . 10 | |
21 | 13, 14, 15, 19, 20 | syl13anc 1218 | . . . . . . . . 9 |
22 | 21 | adantr 274 | . . . . . . . 8 |
23 | 7, 12, 22 | mp2and 429 | . . . . . . 7 |
24 | fimax2gtri.zb | . . . . . . . . . 10 | |
25 | breq2 3933 | . . . . . . . . . . . 12 | |
26 | 25 | notbid 656 | . . . . . . . . . . 11 |
27 | 26 | cbvralv 2654 | . . . . . . . . . 10 |
28 | 24, 27 | sylib 121 | . . . . . . . . 9 |
29 | 28 | r19.21bi 2520 | . . . . . . . 8 |
30 | 29 | ad2antrr 479 | . . . . . . 7 |
31 | 23, 30 | pm2.65da 650 | . . . . . 6 |
32 | 29 | adantr 274 | . . . . . . 7 |
33 | simpr 109 | . . . . . . . . . 10 | |
34 | 33 | iffalsed 3484 | . . . . . . . . 9 |
35 | 34 | breq1d 3939 | . . . . . . . 8 |
36 | 35 | adantlr 468 | . . . . . . 7 |
37 | 32, 36 | mtbird 662 | . . . . . 6 |
38 | exmiddc 821 | . . . . . . . 8 DECID | |
39 | 5, 38 | syl 14 | . . . . . . 7 |
40 | 39 | adantr 274 | . . . . . 6 |
41 | 31, 37, 40 | mpjaodan 787 | . . . . 5 |
42 | 41 | ralrimiva 2505 | . . . 4 |
43 | breq2 3933 | . . . . . 6 | |
44 | 43 | notbid 656 | . . . . 5 |
45 | 44 | cbvralv 2654 | . . . 4 |
46 | 42, 45 | sylib 121 | . . 3 |
47 | 3 | adantr 274 | . . . . . . 7 |
48 | 1 | adantr 274 | . . . . . . 7 |
49 | poirr 4229 | . . . . . . 7 | |
50 | 47, 48, 49 | syl2anc 408 | . . . . . 6 |
51 | 9 | breq1d 3939 | . . . . . 6 |
52 | 50, 51 | mtbird 662 | . . . . 5 |
53 | 34 | breq1d 3939 | . . . . . 6 |
54 | 33, 53 | mtbird 662 | . . . . 5 |
55 | 52, 54, 39 | mpjaodan 787 | . . . 4 |
56 | breq2 3933 | . . . . . . 7 | |
57 | 56 | notbid 656 | . . . . . 6 |
58 | 57 | ralsng 3564 | . . . . 5 |
59 | 1, 58 | syl 14 | . . . 4 |
60 | 55, 59 | mpbird 166 | . . 3 |
61 | ralun 3258 | . . 3 | |
62 | 46, 60, 61 | syl2anc 408 | . 2 |
63 | breq1 3932 | . . . . 5 | |
64 | 63 | notbid 656 | . . . 4 |
65 | 64 | ralbidv 2437 | . . 3 |
66 | 65 | rspcev 2789 | . 2 |
67 | 6, 62, 66 | syl2anc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 DECID wdc 819 w3o 961 wceq 1331 wcel 1480 wne 2308 wral 2416 wrex 2417 cun 3069 wss 3071 c0 3363 cif 3474 csn 3527 class class class wbr 3929 wpo 4216 cfn 6634 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-po 4218 |
This theorem is referenced by: fimax2gtri 6795 |
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