Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > caucvgprlemcl | Unicode version |
Description: Lemma for caucvgpr 7615. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-Sep-2020.) |
Ref | Expression |
---|---|
caucvgpr.f | |
caucvgpr.cau | |
caucvgpr.bnd | |
caucvgpr.lim |
Ref | Expression |
---|---|
caucvgprlemcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgpr.f | . . . 4 | |
2 | caucvgpr.cau | . . . 4 | |
3 | caucvgpr.bnd | . . . . 5 | |
4 | fveq2 5481 | . . . . . . 7 | |
5 | 4 | breq2d 3989 | . . . . . 6 |
6 | 5 | cbvralv 2690 | . . . . 5 |
7 | 3, 6 | sylib 121 | . . . 4 |
8 | caucvgpr.lim | . . . . 5 | |
9 | opeq1 3753 | . . . . . . . . . . . . 13 | |
10 | 9 | eceq1d 6529 | . . . . . . . . . . . 12 |
11 | 10 | fveq2d 5485 | . . . . . . . . . . 11 |
12 | 11 | oveq2d 5853 | . . . . . . . . . 10 |
13 | 12, 4 | breq12d 3990 | . . . . . . . . 9 |
14 | 13 | cbvrexv 2691 | . . . . . . . 8 |
15 | 14 | a1i 9 | . . . . . . 7 |
16 | 15 | rabbiia 2707 | . . . . . 6 |
17 | 4, 11 | oveq12d 5855 | . . . . . . . . . 10 |
18 | 17 | breq1d 3987 | . . . . . . . . 9 |
19 | 18 | cbvrexv 2691 | . . . . . . . 8 |
20 | 19 | a1i 9 | . . . . . . 7 |
21 | 20 | rabbiia 2707 | . . . . . 6 |
22 | 16, 21 | opeq12i 3758 | . . . . 5 |
23 | 8, 22 | eqtri 2185 | . . . 4 |
24 | 1, 2, 7, 23 | caucvgprlemm 7601 | . . 3 |
25 | ssrab2 3223 | . . . . . 6 | |
26 | nqex 7296 | . . . . . . 7 | |
27 | 26 | elpw2 4131 | . . . . . 6 |
28 | 25, 27 | mpbir 145 | . . . . 5 |
29 | ssrab2 3223 | . . . . . 6 | |
30 | 26 | elpw2 4131 | . . . . . 6 |
31 | 29, 30 | mpbir 145 | . . . . 5 |
32 | opelxpi 4631 | . . . . 5 | |
33 | 28, 31, 32 | mp2an 423 | . . . 4 |
34 | 8, 33 | eqeltri 2237 | . . 3 |
35 | 24, 34 | jctil 310 | . 2 |
36 | 1, 2, 7, 23 | caucvgprlemrnd 7606 | . . 3 |
37 | breq1 3980 | . . . . . . 7 | |
38 | fveq2 5481 | . . . . . . . . 9 | |
39 | opeq1 3753 | . . . . . . . . . . . 12 | |
40 | 39 | eceq1d 6529 | . . . . . . . . . . 11 |
41 | 40 | fveq2d 5485 | . . . . . . . . . 10 |
42 | 41 | oveq2d 5853 | . . . . . . . . 9 |
43 | 38, 42 | breq12d 3990 | . . . . . . . 8 |
44 | 38, 41 | oveq12d 5855 | . . . . . . . . 9 |
45 | 44 | breq2d 3989 | . . . . . . . 8 |
46 | 43, 45 | anbi12d 465 | . . . . . . 7 |
47 | 37, 46 | imbi12d 233 | . . . . . 6 |
48 | breq2 3981 | . . . . . . 7 | |
49 | fveq2 5481 | . . . . . . . . . 10 | |
50 | 49 | oveq1d 5852 | . . . . . . . . 9 |
51 | 50 | breq2d 3989 | . . . . . . . 8 |
52 | 49 | breq1d 3987 | . . . . . . . 8 |
53 | 51, 52 | anbi12d 465 | . . . . . . 7 |
54 | 48, 53 | imbi12d 233 | . . . . . 6 |
55 | 47, 54 | cbvral2v 2701 | . . . . 5 |
56 | 2, 55 | sylib 121 | . . . 4 |
57 | 1, 56, 7, 23 | caucvgprlemdisj 7607 | . . 3 |
58 | 1, 2, 7, 23 | caucvgprlemloc 7608 | . . 3 |
59 | 36, 57, 58 | 3jca 1166 | . 2 |
60 | elnp1st2nd 7409 | . 2 | |
61 | 35, 59, 60 | sylanbrc 414 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 w3a 967 wceq 1342 wcel 2135 wral 2442 wrex 2443 crab 2446 wss 3112 cpw 3554 cop 3574 class class class wbr 3977 cxp 4597 wf 5179 cfv 5183 (class class class)co 5837 c1st 6099 c2nd 6100 c1o 6369 cec 6491 cnpi 7205 clti 7208 ceq 7212 cnq 7213 cplq 7215 crq 7217 cltq 7218 cnp 7224 |
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-coll 4092 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-iinf 4560 |
This theorem depends on definitions: df-bi 116 df-dc 825 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-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-tr 4076 df-eprel 4262 df-id 4266 df-po 4269 df-iso 4270 df-iord 4339 df-on 4341 df-suc 4344 df-iom 4563 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-ov 5840 df-oprab 5841 df-mpo 5842 df-1st 6101 df-2nd 6102 df-recs 6265 df-irdg 6330 df-1o 6376 df-oadd 6380 df-omul 6381 df-er 6493 df-ec 6495 df-qs 6499 df-ni 7237 df-pli 7238 df-mi 7239 df-lti 7240 df-plpq 7277 df-mpq 7278 df-enq 7280 df-nqqs 7281 df-plqqs 7282 df-mqqs 7283 df-1nqqs 7284 df-rq 7285 df-ltnqqs 7286 df-inp 7399 |
This theorem is referenced by: caucvgprlemladdfu 7610 caucvgprlemladdrl 7611 caucvgprlem1 7612 caucvgprlem2 7613 caucvgpr 7615 |
Copyright terms: Public domain | W3C validator |