Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > exbtwnzlemstep | Unicode version |
Description: Lemma for exbtwnzlemex 10175. Induction step. (Contributed by Jim Kingdon, 10-May-2022.) |
Ref | Expression |
---|---|
exbtwnzlemstep.k | |
exbtwnzlemstep.a | |
exbtwnzlemstep.tri |
Ref | Expression |
---|---|
exbtwnzlemstep |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpllr 524 | . . . . . . . 8 | |
2 | exbtwnzlemstep.k | . . . . . . . . . 10 | |
3 | 2 | ad3antrrr 484 | . . . . . . . . 9 |
4 | 3 | nnzd 9303 | . . . . . . . 8 |
5 | 1, 4 | zaddcld 9308 | . . . . . . 7 |
6 | simpr 109 | . . . . . . 7 | |
7 | exbtwnzlemstep.a | . . . . . . . . 9 | |
8 | 7 | ad3antrrr 484 | . . . . . . . 8 |
9 | 5 | zred 9304 | . . . . . . . . 9 |
10 | 1red 7905 | . . . . . . . . 9 | |
11 | 9, 10 | readdcld 7919 | . . . . . . . 8 |
12 | 3 | nnred 8861 | . . . . . . . . 9 |
13 | 9, 12 | readdcld 7919 | . . . . . . . 8 |
14 | simplrr 526 | . . . . . . . . 9 | |
15 | 1 | zcnd 9305 | . . . . . . . . . 10 |
16 | 3 | nncnd 8862 | . . . . . . . . . 10 |
17 | 1cnd 7906 | . . . . . . . . . 10 | |
18 | 15, 16, 17 | addassd 7912 | . . . . . . . . 9 |
19 | 14, 18 | breqtrrd 4004 | . . . . . . . 8 |
20 | 3 | nnge1d 8891 | . . . . . . . . 9 |
21 | 10, 12, 9, 20 | leadd2dd 8449 | . . . . . . . 8 |
22 | 8, 11, 13, 19, 21 | ltletrd 8312 | . . . . . . 7 |
23 | breq1 3979 | . . . . . . . . 9 | |
24 | oveq1 5843 | . . . . . . . . . 10 | |
25 | 24 | breq2d 3988 | . . . . . . . . 9 |
26 | 23, 25 | anbi12d 465 | . . . . . . . 8 |
27 | 26 | rspcev 2825 | . . . . . . 7 |
28 | 5, 6, 22, 27 | syl12anc 1225 | . . . . . 6 |
29 | simpllr 524 | . . . . . . 7 | |
30 | simplrl 525 | . . . . . . 7 | |
31 | simpr 109 | . . . . . . 7 | |
32 | breq1 3979 | . . . . . . . . 9 | |
33 | oveq1 5843 | . . . . . . . . . 10 | |
34 | 33 | breq2d 3988 | . . . . . . . . 9 |
35 | 32, 34 | anbi12d 465 | . . . . . . . 8 |
36 | 35 | rspcev 2825 | . . . . . . 7 |
37 | 29, 30, 31, 36 | syl12anc 1225 | . . . . . 6 |
38 | breq1 3979 | . . . . . . . 8 | |
39 | breq2 3980 | . . . . . . . 8 | |
40 | 38, 39 | orbi12d 783 | . . . . . . 7 |
41 | exbtwnzlemstep.tri | . . . . . . . . 9 | |
42 | 41 | ralrimiva 2537 | . . . . . . . 8 |
43 | 42 | ad2antrr 480 | . . . . . . 7 |
44 | simplr 520 | . . . . . . . 8 | |
45 | 2 | ad2antrr 480 | . . . . . . . . 9 |
46 | 45 | nnzd 9303 | . . . . . . . 8 |
47 | 44, 46 | zaddcld 9308 | . . . . . . 7 |
48 | 40, 43, 47 | rspcdva 2830 | . . . . . 6 |
49 | 28, 37, 48 | mpjaodan 788 | . . . . 5 |
50 | 49 | ex 114 | . . . 4 |
51 | 50 | rexlimdva 2581 | . . 3 |
52 | 51 | imp 123 | . 2 |
53 | breq1 3979 | . . . 4 | |
54 | oveq1 5843 | . . . . 5 | |
55 | 54 | breq2d 3988 | . . . 4 |
56 | 53, 55 | anbi12d 465 | . . 3 |
57 | 56 | cbvrexv 2690 | . 2 |
58 | 52, 57 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 698 wceq 1342 wcel 2135 wral 2442 wrex 2443 class class class wbr 3976 (class class class)co 5836 cr 7743 c1 7745 caddc 7747 clt 7924 cle 7925 cn 8848 cz 9182 |
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-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 |
This theorem is referenced by: exbtwnzlemshrink 10174 |
Copyright terms: Public domain | W3C validator |