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Mirrors > Home > ILE Home > Th. List > prodmodclem2 | Unicode version |
Description: Lemma for prodmodc 11352. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 13-Apr-2024.) |
Ref | Expression |
---|---|
prodmo.1 | |
prodmo.2 | |
prodmodc.3 | ♯ |
Ref | Expression |
---|---|
prodmodclem2 | DECID # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 518 | . . . 4 DECID # | |
2 | simplr 519 | . . . 4 DECID # DECID | |
3 | simprr 521 | . . . 4 DECID # | |
4 | 1, 2, 3 | 3jca 1161 | . . 3 DECID # DECID |
5 | 4 | reximi 2529 | . 2 DECID # DECID |
6 | fveq2 5421 | . . . . . 6 | |
7 | 6 | sseq2d 3127 | . . . . 5 |
8 | 6 | raleqdv 2632 | . . . . 5 DECID DECID |
9 | seqeq1 10226 | . . . . . 6 | |
10 | 9 | breq1d 3939 | . . . . 5 |
11 | 7, 8, 10 | 3anbi123d 1290 | . . . 4 DECID DECID |
12 | 11 | cbvrexvw 2659 | . . 3 DECID DECID |
13 | reeanv 2600 | . . . . 5 DECID DECID | |
14 | simprl3 1028 | . . . . . . . . . . . 12 DECID | |
15 | simprl1 1026 | . . . . . . . . . . . . . . 15 DECID | |
16 | uzssz 9350 | . . . . . . . . . . . . . . 15 | |
17 | 15, 16 | sstrdi 3109 | . . . . . . . . . . . . . 14 DECID |
18 | 1zzd 9086 | . . . . . . . . . . . . . . . 16 DECID | |
19 | simplrr 525 | . . . . . . . . . . . . . . . . 17 DECID | |
20 | 19 | nnzd 9177 | . . . . . . . . . . . . . . . 16 DECID |
21 | 18, 20 | fzfigd 10209 | . . . . . . . . . . . . . . 15 DECID |
22 | simprr 521 | . . . . . . . . . . . . . . . . 17 DECID | |
23 | f1oeng 6651 | . . . . . . . . . . . . . . . . 17 | |
24 | 21, 22, 23 | syl2anc 408 | . . . . . . . . . . . . . . . 16 DECID |
25 | 24 | ensymd 6677 | . . . . . . . . . . . . . . 15 DECID |
26 | enfii 6768 | . . . . . . . . . . . . . . 15 | |
27 | 21, 25, 26 | syl2anc 408 | . . . . . . . . . . . . . 14 DECID |
28 | zfz1iso 10589 | . . . . . . . . . . . . . 14 ♯ | |
29 | 17, 27, 28 | syl2anc 408 | . . . . . . . . . . . . 13 DECID ♯ |
30 | prodmo.1 | . . . . . . . . . . . . . . . 16 | |
31 | prodmo.2 | . . . . . . . . . . . . . . . . 17 | |
32 | 31 | ad4ant14 505 | . . . . . . . . . . . . . . . 16 DECID ♯ |
33 | prodmodc.3 | . . . . . . . . . . . . . . . 16 ♯ | |
34 | eqid 2139 | . . . . . . . . . . . . . . . 16 ♯ ♯ | |
35 | simpll2 1021 | . . . . . . . . . . . . . . . . . 18 DECID ♯ DECID | |
36 | 35 | adantl 275 | . . . . . . . . . . . . . . . . 17 DECID ♯ DECID |
37 | eleq1w 2200 | . . . . . . . . . . . . . . . . . . 19 | |
38 | 37 | dcbid 823 | . . . . . . . . . . . . . . . . . 18 DECID DECID |
39 | 38 | rspcv 2785 | . . . . . . . . . . . . . . . . 17 DECID DECID |
40 | 36, 39 | mpan9 279 | . . . . . . . . . . . . . . . 16 DECID ♯ DECID |
41 | simplrr 525 | . . . . . . . . . . . . . . . 16 DECID ♯ | |
42 | simplrl 524 | . . . . . . . . . . . . . . . 16 DECID ♯ | |
43 | 15 | adantrr 470 | . . . . . . . . . . . . . . . 16 DECID ♯ |
44 | simprlr 527 | . . . . . . . . . . . . . . . 16 DECID ♯ | |
45 | simprr 521 | . . . . . . . . . . . . . . . 16 DECID ♯ ♯ | |
46 | 30, 32, 33, 34, 40, 41, 42, 43, 44, 45 | prodmodclem2a 11350 | . . . . . . . . . . . . . . 15 DECID ♯ |
47 | 46 | expr 372 | . . . . . . . . . . . . . 14 DECID ♯ |
48 | 47 | exlimdv 1791 | . . . . . . . . . . . . 13 DECID ♯ |
49 | 29, 48 | mpd 13 | . . . . . . . . . . . 12 DECID |
50 | climuni 11067 | . . . . . . . . . . . 12 | |
51 | 14, 49, 50 | syl2anc 408 | . . . . . . . . . . 11 DECID |
52 | eqeq2 2149 | . . . . . . . . . . 11 | |
53 | 51, 52 | syl5ibrcom 156 | . . . . . . . . . 10 DECID |
54 | 53 | expr 372 | . . . . . . . . 9 DECID |
55 | 54 | impd 252 | . . . . . . . 8 DECID |
56 | 55 | exlimdv 1791 | . . . . . . 7 DECID |
57 | 56 | expimpd 360 | . . . . . 6 DECID |
58 | 57 | rexlimdvva 2557 | . . . . 5 DECID |
59 | 13, 58 | syl5bir 152 | . . . 4 DECID |
60 | 59 | expdimp 257 | . . 3 DECID |
61 | 12, 60 | sylan2b 285 | . 2 DECID |
62 | 5, 61 | sylan2 284 | 1 DECID # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 DECID wdc 819 w3a 962 wceq 1331 wex 1468 wcel 1480 wral 2416 wrex 2417 csb 3003 wss 3071 cif 3474 class class class wbr 3929 cmpt 3989 wf1o 5122 cfv 5123 wiso 5124 (class class class)co 5774 cen 6632 cfn 6634 cc 7623 cc0 7625 c1 7626 cmul 7630 clt 7805 cle 7806 # cap 8348 cn 8725 cz 9059 cuz 9331 cfz 9795 cseq 10223 ♯chash 10526 cli 11052 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7716 ax-resscn 7717 ax-1cn 7718 ax-1re 7719 ax-icn 7720 ax-addcl 7721 ax-addrcl 7722 ax-mulcl 7723 ax-mulrcl 7724 ax-addcom 7725 ax-mulcom 7726 ax-addass 7727 ax-mulass 7728 ax-distr 7729 ax-i2m1 7730 ax-0lt1 7731 ax-1rid 7732 ax-0id 7733 ax-rnegex 7734 ax-precex 7735 ax-cnre 7736 ax-pre-ltirr 7737 ax-pre-ltwlin 7738 ax-pre-lttrn 7739 ax-pre-apti 7740 ax-pre-ltadd 7741 ax-pre-mulgt0 7742 ax-pre-mulext 7743 ax-arch 7744 ax-caucvg 7745 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-pnf 7807 df-mnf 7808 df-xr 7809 df-ltxr 7810 df-le 7811 df-sub 7940 df-neg 7941 df-reap 8342 df-ap 8349 df-div 8438 df-inn 8726 df-2 8784 df-3 8785 df-4 8786 df-n0 8983 df-z 9060 df-uz 9332 df-q 9417 df-rp 9447 df-fz 9796 df-fzo 9925 df-seqfrec 10224 df-exp 10298 df-ihash 10527 df-cj 10619 df-re 10620 df-im 10621 df-rsqrt 10775 df-abs 10776 df-clim 11053 |
This theorem is referenced by: prodmodc 11352 |
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