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Mirrors > Home > ILE Home > Th. List > fimax2gtrilemstep | Unicode version |
Description: Lemma for fimax2gtri 6763. 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 6761 | . . 3 DECID |
6 | 1, 2, 5 | ifcldcd 3477 | . 2 |
7 | simplr 504 | . . . . . . . 8 | |
8 | simpr 109 | . . . . . . . . . . . 12 | |
9 | 8 | iftrued 3451 | . . . . . . . . . . 11 |
10 | 9 | breq1d 3909 | . . . . . . . . . 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 504 | . . . . . . . . . . 11 | |
19 | 17, 18 | sseldd 3068 | . . . . . . . . . 10 |
20 | potr 4200 | . . . . . . . . . 10 | |
21 | 13, 14, 15, 19, 20 | syl13anc 1203 | . . . . . . . . 9 |
22 | 21 | adantr 274 | . . . . . . . 8 |
23 | 7, 12, 22 | mp2and 429 | . . . . . . 7 |
24 | fimax2gtri.zb | . . . . . . . . . 10 | |
25 | breq2 3903 | . . . . . . . . . . . 12 | |
26 | 25 | notbid 641 | . . . . . . . . . . 11 |
27 | 26 | cbvralv 2631 | . . . . . . . . . 10 |
28 | 24, 27 | sylib 121 | . . . . . . . . 9 |
29 | 28 | r19.21bi 2497 | . . . . . . . 8 |
30 | 29 | ad2antrr 479 | . . . . . . 7 |
31 | 23, 30 | pm2.65da 635 | . . . . . 6 |
32 | 29 | adantr 274 | . . . . . . 7 |
33 | simpr 109 | . . . . . . . . . 10 | |
34 | 33 | iffalsed 3454 | . . . . . . . . 9 |
35 | 34 | breq1d 3909 | . . . . . . . 8 |
36 | 35 | adantlr 468 | . . . . . . 7 |
37 | 32, 36 | mtbird 647 | . . . . . 6 |
38 | exmiddc 806 | . . . . . . . 8 DECID | |
39 | 5, 38 | syl 14 | . . . . . . 7 |
40 | 39 | adantr 274 | . . . . . 6 |
41 | 31, 37, 40 | mpjaodan 772 | . . . . 5 |
42 | 41 | ralrimiva 2482 | . . . 4 |
43 | breq2 3903 | . . . . . 6 | |
44 | 43 | notbid 641 | . . . . 5 |
45 | 44 | cbvralv 2631 | . . . 4 |
46 | 42, 45 | sylib 121 | . . 3 |
47 | 3 | adantr 274 | . . . . . . 7 |
48 | 1 | adantr 274 | . . . . . . 7 |
49 | poirr 4199 | . . . . . . 7 | |
50 | 47, 48, 49 | syl2anc 408 | . . . . . 6 |
51 | 9 | breq1d 3909 | . . . . . 6 |
52 | 50, 51 | mtbird 647 | . . . . 5 |
53 | 34 | breq1d 3909 | . . . . . 6 |
54 | 33, 53 | mtbird 647 | . . . . 5 |
55 | 52, 54, 39 | mpjaodan 772 | . . . 4 |
56 | breq2 3903 | . . . . . . 7 | |
57 | 56 | notbid 641 | . . . . . 6 |
58 | 57 | ralsng 3534 | . . . . 5 |
59 | 1, 58 | syl 14 | . . . 4 |
60 | 55, 59 | mpbird 166 | . . 3 |
61 | ralun 3228 | . . 3 | |
62 | 46, 60, 61 | syl2anc 408 | . 2 |
63 | breq1 3902 | . . . . 5 | |
64 | 63 | notbid 641 | . . . 4 |
65 | 64 | ralbidv 2414 | . . 3 |
66 | 65 | rspcev 2763 | . 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 682 DECID wdc 804 w3o 946 wceq 1316 wcel 1465 wne 2285 wral 2393 wrex 2394 cun 3039 wss 3041 c0 3333 cif 3444 csn 3497 class class class wbr 3899 wpo 4186 cfn 6602 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-if 3445 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-po 4188 |
This theorem is referenced by: fimax2gtri 6763 |
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