Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > enumctlemm | Unicode version |
Description: Lemma for enumct 7000. The case where is greater than zero. (Contributed by Jim Kingdon, 13-Mar-2023.) |
Ref | Expression |
---|---|
enumctlemm.f | |
enumctlemm.n | |
enumctlemm.n0 | |
enumctlemm.g |
Ref | Expression |
---|---|
enumctlemm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enumctlemm.f | . . . . . . 7 | |
2 | fof 5345 | . . . . . . 7 | |
3 | 1, 2 | syl 14 | . . . . . 6 |
4 | 3 | ffvelrnda 5555 | . . . . 5 |
5 | 4 | adantlr 468 | . . . 4 |
6 | enumctlemm.n0 | . . . . . 6 | |
7 | 3, 6 | ffvelrnd 5556 | . . . . 5 |
8 | 7 | ad2antrr 479 | . . . 4 |
9 | simpr 109 | . . . . 5 | |
10 | enumctlemm.n | . . . . . 6 | |
11 | 10 | adantr 274 | . . . . 5 |
12 | nndcel 6396 | . . . . 5 DECID | |
13 | 9, 11, 12 | syl2anc 408 | . . . 4 DECID |
14 | 5, 8, 13 | ifcldadc 3501 | . . 3 |
15 | enumctlemm.g | . . 3 | |
16 | 14, 15 | fmptd 5574 | . 2 |
17 | foelrn 5654 | . . . . . 6 | |
18 | 1, 17 | sylan 281 | . . . . 5 |
19 | eleq1w 2200 | . . . . . . . . . . 11 | |
20 | fveq2 5421 | . . . . . . . . . . 11 | |
21 | 19, 20 | ifbieq1d 3494 | . . . . . . . . . 10 |
22 | simpr 109 | . . . . . . . . . . 11 | |
23 | 10 | adantr 274 | . . . . . . . . . . 11 |
24 | elnn 4519 | . . . . . . . . . . 11 | |
25 | 22, 23, 24 | syl2anc 408 | . . . . . . . . . 10 |
26 | 22 | iftrued 3481 | . . . . . . . . . . 11 |
27 | 3 | ffvelrnda 5555 | . . . . . . . . . . 11 |
28 | 26, 27 | eqeltrd 2216 | . . . . . . . . . 10 |
29 | 15, 21, 25, 28 | fvmptd3 5514 | . . . . . . . . 9 |
30 | 29, 26 | eqtrd 2172 | . . . . . . . 8 |
31 | 30 | eqeq2d 2151 | . . . . . . 7 |
32 | 31 | rexbidva 2434 | . . . . . 6 |
33 | 32 | adantr 274 | . . . . 5 |
34 | 18, 33 | mpbird 166 | . . . 4 |
35 | omelon 4522 | . . . . . . 7 | |
36 | 35 | onelssi 4351 | . . . . . 6 |
37 | ssrexv 3162 | . . . . . 6 | |
38 | 10, 36, 37 | 3syl 17 | . . . . 5 |
39 | 38 | adantr 274 | . . . 4 |
40 | 34, 39 | mpd 13 | . . 3 |
41 | 40 | ralrimiva 2505 | . 2 |
42 | dffo3 5567 | . 2 | |
43 | 16, 41, 42 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 DECID wdc 819 wceq 1331 wcel 1480 wral 2416 wrex 2417 wss 3071 c0 3363 cif 3474 cmpt 3989 com 4504 wf 5119 wfo 5121 cfv 5123 |
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-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 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-ral 2421 df-rex 2422 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-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 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-fo 5129 df-fv 5131 |
This theorem is referenced by: enumct 7000 |
Copyright terms: Public domain | W3C validator |