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Mirrors > Home > ILE Home > Th. List > zsupcllemstep | Unicode version |
Description: Lemma for zsupcl 11552. Induction step. (Contributed by Jim Kingdon, 7-Dec-2021.) |
Ref | Expression |
---|---|
zsupcllemstep.dc | DECID |
Ref | Expression |
---|---|
zsupcllemstep |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz 9291 | . . . . 5 | |
2 | 1 | ad3antrrr 483 | . . . 4 |
3 | nfv 1493 | . . . . . . . 8 | |
4 | nfv 1493 | . . . . . . . . 9 | |
5 | nfcv 2258 | . . . . . . . . . 10 | |
6 | nfra1 2443 | . . . . . . . . . . 11 | |
7 | nfra1 2443 | . . . . . . . . . . 11 | |
8 | 6, 7 | nfan 1529 | . . . . . . . . . 10 |
9 | 5, 8 | nfrexya 2451 | . . . . . . . . 9 |
10 | 4, 9 | nfim 1536 | . . . . . . . 8 |
11 | 3, 10 | nfan 1529 | . . . . . . 7 |
12 | nfv 1493 | . . . . . . 7 | |
13 | 11, 12 | nfan 1529 | . . . . . 6 |
14 | nfv 1493 | . . . . . 6 | |
15 | 13, 14 | nfan 1529 | . . . . 5 |
16 | nfcv 2258 | . . . . . . . . . . 11 | |
17 | 16 | elrabsf 2919 | . . . . . . . . . 10 |
18 | 17 | simprbi 273 | . . . . . . . . 9 |
19 | sbsbc 2886 | . . . . . . . . 9 | |
20 | 18, 19 | sylibr 133 | . . . . . . . 8 |
21 | 20 | ad2antlr 480 | . . . . . . 7 |
22 | elrabi 2810 | . . . . . . . . . . 11 | |
23 | zltp1le 9066 | . . . . . . . . . . 11 | |
24 | 2, 22, 23 | syl2an 287 | . . . . . . . . . 10 |
25 | 24 | biimpa 294 | . . . . . . . . 9 |
26 | 2 | peano2zd 9134 | . . . . . . . . . . 11 |
27 | eluz 9295 | . . . . . . . . . . 11 | |
28 | 26, 22, 27 | syl2an 287 | . . . . . . . . . 10 |
29 | 28 | adantr 274 | . . . . . . . . 9 |
30 | 25, 29 | mpbird 166 | . . . . . . . 8 |
31 | simprr 506 | . . . . . . . . 9 | |
32 | 31 | ad3antrrr 483 | . . . . . . . 8 |
33 | nfs1v 1892 | . . . . . . . . . 10 | |
34 | 33 | nfn 1621 | . . . . . . . . 9 |
35 | sbequ12 1729 | . . . . . . . . . 10 | |
36 | 35 | notbid 641 | . . . . . . . . 9 |
37 | 34, 36 | rspc 2757 | . . . . . . . 8 |
38 | 30, 32, 37 | sylc 62 | . . . . . . 7 |
39 | 21, 38 | pm2.65da 635 | . . . . . 6 |
40 | 39 | ex 114 | . . . . 5 |
41 | 15, 40 | ralrimi 2480 | . . . 4 |
42 | 2 | ad2antrr 479 | . . . . . . . 8 |
43 | simpllr 508 | . . . . . . . 8 | |
44 | 16 | elrabsf 2919 | . . . . . . . 8 |
45 | 42, 43, 44 | sylanbrc 413 | . . . . . . 7 |
46 | breq2 3903 | . . . . . . . 8 | |
47 | 46 | rspcev 2763 | . . . . . . 7 |
48 | 45, 47 | sylancom 416 | . . . . . 6 |
49 | 48 | exp31 361 | . . . . 5 |
50 | 15, 49 | ralrimi 2480 | . . . 4 |
51 | breq1 3902 | . . . . . . . 8 | |
52 | 51 | notbid 641 | . . . . . . 7 |
53 | 52 | ralbidv 2414 | . . . . . 6 |
54 | breq2 3903 | . . . . . . . 8 | |
55 | 54 | imbi1d 230 | . . . . . . 7 |
56 | 55 | ralbidv 2414 | . . . . . 6 |
57 | 53, 56 | anbi12d 464 | . . . . 5 |
58 | 57 | rspcev 2763 | . . . 4 |
59 | 2, 41, 50, 58 | syl12anc 1199 | . . 3 |
60 | sbcng 2921 | . . . . . . . 8 | |
61 | 60 | ad2antrr 479 | . . . . . . 7 |
62 | 61 | biimpar 295 | . . . . . 6 |
63 | sbcsng 3552 | . . . . . . 7 | |
64 | 63 | ad3antrrr 483 | . . . . . 6 |
65 | 62, 64 | mpbid 146 | . . . . 5 |
66 | simplrr 510 | . . . . 5 | |
67 | uzid 9296 | . . . . . . . . . . 11 | |
68 | peano2uz 9334 | . . . . . . . . . . 11 | |
69 | 67, 68 | syl 14 | . . . . . . . . . 10 |
70 | fzouzsplit 9911 | . . . . . . . . . 10 ..^ | |
71 | 1, 69, 70 | 3syl 17 | . . . . . . . . 9 ..^ |
72 | fzosn 9937 | . . . . . . . . . . 11 ..^ | |
73 | 1, 72 | syl 14 | . . . . . . . . . 10 ..^ |
74 | 73 | uneq1d 3199 | . . . . . . . . 9 ..^ |
75 | 71, 74 | eqtrd 2150 | . . . . . . . 8 |
76 | 75 | raleqdv 2609 | . . . . . . 7 |
77 | ralunb 3227 | . . . . . . 7 | |
78 | 76, 77 | syl6bb 195 | . . . . . 6 |
79 | 78 | ad3antrrr 483 | . . . . 5 |
80 | 65, 66, 79 | mpbir2and 913 | . . . 4 |
81 | simprl 505 | . . . . . 6 | |
82 | simplr 504 | . . . . . 6 | |
83 | 81, 82 | mpand 425 | . . . . 5 |
84 | 83 | adantr 274 | . . . 4 |
85 | 80, 84 | mpd 13 | . . 3 |
86 | zsupcllemstep.dc | . . . . . . 7 DECID | |
87 | 86 | ralrimiva 2482 | . . . . . 6 DECID |
88 | 81, 87 | syl 14 | . . . . 5 DECID |
89 | nfsbc1v 2900 | . . . . . . . 8 | |
90 | 89 | nfdc 1622 | . . . . . . 7 DECID |
91 | sbceq1a 2891 | . . . . . . . 8 | |
92 | 91 | dcbid 808 | . . . . . . 7 DECID DECID |
93 | 90, 92 | rspc 2757 | . . . . . 6 DECID DECID |
94 | 93 | ad2antrr 479 | . . . . 5 DECID DECID |
95 | 88, 94 | mpd 13 | . . . 4 DECID |
96 | exmiddc 806 | . . . 4 DECID | |
97 | 95, 96 | syl 14 | . . 3 |
98 | 59, 85, 97 | mpjaodan 772 | . 2 |
99 | 98 | exp31 361 | 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 wsb 1720 wral 2393 wrex 2394 crab 2397 wsbc 2882 cun 3039 csn 3497 class class class wbr 3899 cfv 5093 (class class class)co 5742 cr 7587 c1 7589 caddc 7591 clt 7768 cle 7769 cz 9012 cuz 9282 ..^cfzo 9874 |
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-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-csb 2976 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-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8685 df-n0 8936 df-z 9013 df-uz 9283 df-fz 9746 df-fzo 9875 |
This theorem is referenced by: zsupcllemex 11551 |
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