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Mirrors > Home > ILE Home > Th. List > recvguniq | Unicode version |
Description: Limits are unique. (Contributed by Jim Kingdon, 7-Aug-2021.) |
Ref | Expression |
---|---|
recvguniq.f | |
recvguniq.lre | |
recvguniq.l | |
recvguniq.mre | |
recvguniq.m |
Ref | Expression |
---|---|
recvguniq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recvguniq.lre | . . . . 5 | |
2 | recvguniq.mre | . . . . 5 | |
3 | reaplt 8317 | . . . . 5 # | |
4 | 1, 2, 3 | syl2anc 408 | . . . 4 # |
5 | oveq2 5750 | . . . . . . . . . . . 12 | |
6 | 5 | breq2d 3911 | . . . . . . . . . . 11 |
7 | oveq2 5750 | . . . . . . . . . . . 12 | |
8 | 7 | breq2d 3911 | . . . . . . . . . . 11 |
9 | 6, 8 | anbi12d 464 | . . . . . . . . . 10 |
10 | oveq2 5750 | . . . . . . . . . . . 12 | |
11 | 10 | breq2d 3911 | . . . . . . . . . . 11 |
12 | 7 | breq2d 3911 | . . . . . . . . . . 11 |
13 | 11, 12 | anbi12d 464 | . . . . . . . . . 10 |
14 | 9, 13 | anbi12d 464 | . . . . . . . . 9 |
15 | 14 | rexbidv 2415 | . . . . . . . 8 |
16 | recvguniq.l | . . . . . . . . . . . 12 | |
17 | recvguniq.m | . . . . . . . . . . . 12 | |
18 | r19.26 2535 | . . . . . . . . . . . 12 | |
19 | 16, 17, 18 | sylanbrc 413 | . . . . . . . . . . 11 |
20 | nnuz 9317 | . . . . . . . . . . . . 13 | |
21 | 20 | rexanuz2 10718 | . . . . . . . . . . . 12 |
22 | 21 | ralbii 2418 | . . . . . . . . . . 11 |
23 | 19, 22 | sylibr 133 | . . . . . . . . . 10 |
24 | 20 | r19.2uz 10720 | . . . . . . . . . . 11 |
25 | 24 | ralimi 2472 | . . . . . . . . . 10 |
26 | 23, 25 | syl 14 | . . . . . . . . 9 |
27 | 26 | adantr 274 | . . . . . . . 8 |
28 | simpr 109 | . . . . . . . . . 10 | |
29 | 1 | adantr 274 | . . . . . . . . . . 11 |
30 | 2 | adantr 274 | . . . . . . . . . . 11 |
31 | difrp 9435 | . . . . . . . . . . 11 | |
32 | 29, 30, 31 | syl2anc 408 | . . . . . . . . . 10 |
33 | 28, 32 | mpbid 146 | . . . . . . . . 9 |
34 | 33 | rphalfcld 9451 | . . . . . . . 8 |
35 | 15, 27, 34 | rspcdva 2768 | . . . . . . 7 |
36 | recvguniq.f | . . . . . . . . 9 | |
37 | 36 | ad2antrr 479 | . . . . . . . 8 |
38 | 2 | ad2antrr 479 | . . . . . . . 8 |
39 | 1 | ad2antrr 479 | . . . . . . . 8 |
40 | simprl 505 | . . . . . . . 8 | |
41 | simprrr 514 | . . . . . . . . 9 | |
42 | 41 | adantl 275 | . . . . . . . 8 |
43 | simprll 511 | . . . . . . . . 9 | |
44 | 43 | adantl 275 | . . . . . . . 8 |
45 | 37, 38, 39, 40, 42, 44 | recvguniqlem 10721 | . . . . . . 7 |
46 | 35, 45 | rexlimddv 2531 | . . . . . 6 |
47 | 46 | ex 114 | . . . . 5 |
48 | oveq2 5750 | . . . . . . . . . . . 12 | |
49 | 48 | breq2d 3911 | . . . . . . . . . . 11 |
50 | oveq2 5750 | . . . . . . . . . . . 12 | |
51 | 50 | breq2d 3911 | . . . . . . . . . . 11 |
52 | 49, 51 | anbi12d 464 | . . . . . . . . . 10 |
53 | oveq2 5750 | . . . . . . . . . . . 12 | |
54 | 53 | breq2d 3911 | . . . . . . . . . . 11 |
55 | 50 | breq2d 3911 | . . . . . . . . . . 11 |
56 | 54, 55 | anbi12d 464 | . . . . . . . . . 10 |
57 | 52, 56 | anbi12d 464 | . . . . . . . . 9 |
58 | 57 | rexbidv 2415 | . . . . . . . 8 |
59 | 26 | adantr 274 | . . . . . . . 8 |
60 | difrp 9435 | . . . . . . . . . . 11 | |
61 | 2, 1, 60 | syl2anc 408 | . . . . . . . . . 10 |
62 | 61 | biimpa 294 | . . . . . . . . 9 |
63 | 62 | rphalfcld 9451 | . . . . . . . 8 |
64 | 58, 59, 63 | rspcdva 2768 | . . . . . . 7 |
65 | 36 | ad2antrr 479 | . . . . . . . 8 |
66 | 1 | ad2antrr 479 | . . . . . . . 8 |
67 | 2 | ad2antrr 479 | . . . . . . . 8 |
68 | simprl 505 | . . . . . . . 8 | |
69 | simprlr 512 | . . . . . . . . 9 | |
70 | 69 | adantl 275 | . . . . . . . 8 |
71 | simprrl 513 | . . . . . . . . 9 | |
72 | 71 | adantl 275 | . . . . . . . 8 |
73 | 65, 66, 67, 68, 70, 72 | recvguniqlem 10721 | . . . . . . 7 |
74 | 64, 73 | rexlimddv 2531 | . . . . . 6 |
75 | 74 | ex 114 | . . . . 5 |
76 | 47, 75 | jaod 691 | . . . 4 |
77 | 4, 76 | sylbid 149 | . . 3 # |
78 | dfnot 1334 | . . 3 # # | |
79 | 77, 78 | sylibr 133 | . 2 # |
80 | 1 | recnd 7762 | . . 3 |
81 | 2 | recnd 7762 | . . 3 |
82 | apti 8351 | . . 3 # | |
83 | 80, 81, 82 | syl2anc 408 | . 2 # |
84 | 79, 83 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 wceq 1316 wfal 1321 wcel 1465 wral 2393 wrex 2394 class class class wbr 3899 wf 5089 cfv 5093 (class class class)co 5742 cc 7586 cr 7587 c1 7589 caddc 7591 clt 7768 cmin 7901 # cap 8310 cdiv 8399 cn 8684 c2 8735 cuz 9282 crp 9397 |
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-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
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-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-if 3445 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-mpt 3961 df-id 4185 df-po 4188 df-iso 4189 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-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8304 df-ap 8311 df-div 8400 df-inn 8685 df-2 8743 df-n0 8936 df-z 9013 df-uz 9283 df-rp 9398 |
This theorem is referenced by: resqrexlemsqa 10751 |
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