Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > algcvgblem | Unicode version |
Description: Lemma for algcvgb 11658. (Contributed by Paul Chapman, 31-Mar-2011.) |
Ref | Expression |
---|---|
algcvgblem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0z 9042 | . . . . . . . . 9 | |
2 | 0z 9033 | . . . . . . . . 9 | |
3 | zdceq 9094 | . . . . . . . . 9 DECID | |
4 | 1, 2, 3 | sylancl 409 | . . . . . . . 8 DECID |
5 | 4 | dcned 2291 | . . . . . . 7 DECID |
6 | imordc 867 | . . . . . . 7 DECID | |
7 | 5, 6 | syl 14 | . . . . . 6 |
8 | 7 | adantl 275 | . . . . 5 |
9 | nn0z 9042 | . . . . . . . . . . . . . 14 | |
10 | zltnle 9068 | . . . . . . . . . . . . . 14 | |
11 | 2, 9, 10 | sylancr 410 | . . . . . . . . . . . . 13 |
12 | 11 | adantr 274 | . . . . . . . . . . . 12 |
13 | nn0le0eq0 8973 | . . . . . . . . . . . . . 14 | |
14 | 13 | notbid 641 | . . . . . . . . . . . . 13 |
15 | 14 | adantr 274 | . . . . . . . . . . . 12 |
16 | 12, 15 | bitrd 187 | . . . . . . . . . . 11 |
17 | df-ne 2286 | . . . . . . . . . . 11 | |
18 | 16, 17 | syl6bbr 197 | . . . . . . . . . 10 |
19 | 18 | anbi2d 459 | . . . . . . . . 9 |
20 | 1 | adantl 275 | . . . . . . . . . . . . . 14 |
21 | 20, 2, 3 | sylancl 409 | . . . . . . . . . . . . 13 DECID |
22 | nnedc 2290 | . . . . . . . . . . . . 13 DECID | |
23 | 21, 22 | syl 14 | . . . . . . . . . . . 12 |
24 | breq1 3902 | . . . . . . . . . . . 12 | |
25 | 23, 24 | syl6bi 162 | . . . . . . . . . . 11 |
26 | bi2 129 | . . . . . . . . . . 11 | |
27 | 25, 26 | syl6 33 | . . . . . . . . . 10 |
28 | 27 | impd 252 | . . . . . . . . 9 |
29 | 19, 28 | sylbird 169 | . . . . . . . 8 |
30 | 29 | expd 256 | . . . . . . 7 |
31 | ax-1 6 | . . . . . . 7 | |
32 | 30, 31 | jctir 311 | . . . . . 6 |
33 | jaob 684 | . . . . . 6 | |
34 | 32, 33 | sylibr 133 | . . . . 5 |
35 | 8, 34 | sylbid 149 | . . . 4 |
36 | nn0ge0 8970 | . . . . . . . 8 | |
37 | 36 | adantl 275 | . . . . . . 7 |
38 | nn0re 8954 | . . . . . . . 8 | |
39 | nn0re 8954 | . . . . . . . 8 | |
40 | 0re 7734 | . . . . . . . . 9 | |
41 | lelttr 7820 | . . . . . . . . 9 | |
42 | 40, 41 | mp3an1 1287 | . . . . . . . 8 |
43 | 38, 39, 42 | syl2anr 288 | . . . . . . 7 |
44 | 37, 43 | mpand 425 | . . . . . 6 |
45 | 44, 18 | sylibd 148 | . . . . 5 |
46 | 45 | imim2d 54 | . . . 4 |
47 | 35, 46 | jcad 305 | . . 3 |
48 | pm3.34 343 | . . 3 | |
49 | 47, 48 | impbid1 141 | . 2 |
50 | con34bdc 841 | . . . . 5 DECID | |
51 | 21, 50 | syl 14 | . . . 4 |
52 | df-ne 2286 | . . . . 5 | |
53 | 52, 17 | imbi12i 238 | . . . 4 |
54 | 51, 53 | syl6bbr 197 | . . 3 |
55 | 54 | anbi2d 459 | . 2 |
56 | 49, 55 | bitr4d 190 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 DECID wdc 804 wceq 1316 wcel 1465 wne 2285 class class class wbr 3899 cr 7587 cc0 7588 clt 7768 cle 7769 cn0 8945 cz 9022 |
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-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-stab 801 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 |
This theorem is referenced by: algcvgb 11658 |
Copyright terms: Public domain | W3C validator |