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Mirrors > Home > ILE Home > Th. List > addcncntoplem | Unicode version |
Description: Lemma for addcncntop 12994, subcncntop 12995, and mulcncntop 12996. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 22-Oct-2023.) |
Ref | Expression |
---|---|
addcncntop.j | |
addcn.2 | |
addcn.3 |
Ref | Expression |
---|---|
addcncntoplem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcn.2 | . 2 | |
2 | addcn.3 | . . . . 5 | |
3 | 2 | 3coml 1192 | . . . 4 |
4 | rpmincl 11141 | . . . . . . 7 inf | |
5 | 4 | adantl 275 | . . . . . 6 inf |
6 | simpll1 1021 | . . . . . . . . . . . . 13 | |
7 | simprl 521 | . . . . . . . . . . . . 13 | |
8 | eqid 2157 | . . . . . . . . . . . . . . 15 | |
9 | 8 | cnmetdval 12971 | . . . . . . . . . . . . . 14 |
10 | abssub 11005 | . . . . . . . . . . . . . 14 | |
11 | 9, 10 | eqtrd 2190 | . . . . . . . . . . . . 13 |
12 | 6, 7, 11 | syl2anc 409 | . . . . . . . . . . . 12 |
13 | 12 | breq1d 3976 | . . . . . . . . . . 11 inf inf |
14 | 7, 6 | subcld 8187 | . . . . . . . . . . . . 13 |
15 | 14 | abscld 11085 | . . . . . . . . . . . 12 |
16 | simplrl 525 | . . . . . . . . . . . . 13 | |
17 | 16 | rpred 9604 | . . . . . . . . . . . 12 |
18 | simplrr 526 | . . . . . . . . . . . . 13 | |
19 | 18 | rpred 9604 | . . . . . . . . . . . 12 |
20 | ltmininf 11138 | . . . . . . . . . . . 12 inf | |
21 | 15, 17, 19, 20 | syl3anc 1220 | . . . . . . . . . . 11 inf |
22 | 13, 21 | bitrd 187 | . . . . . . . . . 10 inf |
23 | simpl 108 | . . . . . . . . . 10 | |
24 | 22, 23 | syl6bi 162 | . . . . . . . . 9 inf |
25 | simpll2 1022 | . . . . . . . . . . . . 13 | |
26 | simprr 522 | . . . . . . . . . . . . 13 | |
27 | 8 | cnmetdval 12971 | . . . . . . . . . . . . . 14 |
28 | abssub 11005 | . . . . . . . . . . . . . 14 | |
29 | 27, 28 | eqtrd 2190 | . . . . . . . . . . . . 13 |
30 | 25, 26, 29 | syl2anc 409 | . . . . . . . . . . . 12 |
31 | 30 | breq1d 3976 | . . . . . . . . . . 11 inf inf |
32 | 26, 25 | subcld 8187 | . . . . . . . . . . . . 13 |
33 | 32 | abscld 11085 | . . . . . . . . . . . 12 |
34 | ltmininf 11138 | . . . . . . . . . . . 12 inf | |
35 | 33, 17, 19, 34 | syl3anc 1220 | . . . . . . . . . . 11 inf |
36 | 31, 35 | bitrd 187 | . . . . . . . . . 10 inf |
37 | simpr 109 | . . . . . . . . . 10 | |
38 | 36, 37 | syl6bi 162 | . . . . . . . . 9 inf |
39 | 24, 38 | anim12d 333 | . . . . . . . 8 inf inf |
40 | 1 | fovcl 5927 | . . . . . . . . . . . 12 |
41 | 6, 25, 40 | syl2anc 409 | . . . . . . . . . . 11 |
42 | 1 | fovcl 5927 | . . . . . . . . . . . 12 |
43 | 42 | adantl 275 | . . . . . . . . . . 11 |
44 | 8 | cnmetdval 12971 | . . . . . . . . . . . 12 |
45 | abssub 11005 | . . . . . . . . . . . 12 | |
46 | 44, 45 | eqtrd 2190 | . . . . . . . . . . 11 |
47 | 41, 43, 46 | syl2anc 409 | . . . . . . . . . 10 |
48 | 47 | breq1d 3976 | . . . . . . . . 9 |
49 | 48 | biimprd 157 | . . . . . . . 8 |
50 | 39, 49 | imim12d 74 | . . . . . . 7 inf inf |
51 | 50 | ralimdvva 2526 | . . . . . 6 inf inf |
52 | breq2 3970 | . . . . . . . . . 10 inf inf | |
53 | breq2 3970 | . . . . . . . . . 10 inf inf | |
54 | 52, 53 | anbi12d 465 | . . . . . . . . 9 inf inf inf |
55 | 54 | imbi1d 230 | . . . . . . . 8 inf inf inf |
56 | 55 | 2ralbidv 2481 | . . . . . . 7 inf inf inf |
57 | 56 | rspcev 2816 | . . . . . 6 inf inf inf |
58 | 5, 51, 57 | syl6an 1414 | . . . . 5 |
59 | 58 | rexlimdvva 2582 | . . . 4 |
60 | 3, 59 | mpd 13 | . . 3 |
61 | 60 | rgen3 2544 | . 2 |
62 | cnxmet 12973 | . . 3 | |
63 | addcncntop.j | . . . 4 | |
64 | 63, 63, 63 | txmetcn 12961 | . . 3 |
65 | 62, 62, 62, 64 | mp3an 1319 | . 2 |
66 | 1, 61, 65 | mpbir2an 927 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wceq 1335 wcel 2128 wral 2435 wrex 2436 cpr 3561 class class class wbr 3966 cxp 4585 ccom 4591 wf 5167 cfv 5171 (class class class)co 5825 infcinf 6928 cc 7731 cr 7732 clt 7913 cmin 8047 crp 9561 cabs 10901 cxmet 12422 cmopn 12427 ccn 12627 ctx 12694 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-mulrcl 7832 ax-addcom 7833 ax-mulcom 7834 ax-addass 7835 ax-mulass 7836 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-1rid 7840 ax-0id 7841 ax-rnegex 7842 ax-precex 7843 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-apti 7848 ax-pre-ltadd 7849 ax-pre-mulgt0 7850 ax-pre-mulext 7851 ax-arch 7852 ax-caucvg 7853 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-po 4257 df-iso 4258 df-iord 4327 df-on 4329 df-ilim 4330 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-isom 5180 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-frec 6339 df-map 6596 df-sup 6929 df-inf 6930 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-reap 8451 df-ap 8458 df-div 8547 df-inn 8835 df-2 8893 df-3 8894 df-4 8895 df-n0 9092 df-z 9169 df-uz 9441 df-q 9530 df-rp 9562 df-xneg 9680 df-xadd 9681 df-seqfrec 10349 df-exp 10423 df-cj 10746 df-re 10747 df-im 10748 df-rsqrt 10902 df-abs 10903 df-topgen 12414 df-psmet 12429 df-xmet 12430 df-met 12431 df-bl 12432 df-mopn 12433 df-top 12438 df-topon 12451 df-bases 12483 df-cn 12630 df-cnp 12631 df-tx 12695 |
This theorem is referenced by: addcncntop 12994 subcncntop 12995 mulcncntop 12996 |
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