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Mirrors > Home > ILE Home > Th. List > enumctlemm | Unicode version |
Description: Lemma for enumct 6993. 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 5340 | . . . . . . 7 | |
3 | 1, 2 | syl 14 | . . . . . 6 |
4 | 3 | ffvelrnda 5548 | . . . . 5 |
5 | 4 | adantlr 468 | . . . 4 |
6 | enumctlemm.n0 | . . . . . 6 | |
7 | 3, 6 | ffvelrnd 5549 | . . . . 5 |
8 | 7 | ad2antrr 479 | . . . 4 |
9 | simpr 109 | . . . . 5 | |
10 | enumctlemm.n | . . . . . 6 | |
11 | 10 | adantr 274 | . . . . 5 |
12 | nndcel 6389 | . . . . 5 DECID | |
13 | 9, 11, 12 | syl2anc 408 | . . . 4 DECID |
14 | 5, 8, 13 | ifcldadc 3496 | . . 3 |
15 | enumctlemm.g | . . 3 | |
16 | 14, 15 | fmptd 5567 | . 2 |
17 | foelrn 5647 | . . . . . 6 | |
18 | 1, 17 | sylan 281 | . . . . 5 |
19 | eleq1w 2198 | . . . . . . . . . . 11 | |
20 | fveq2 5414 | . . . . . . . . . . 11 | |
21 | 19, 20 | ifbieq1d 3489 | . . . . . . . . . 10 |
22 | simpr 109 | . . . . . . . . . . 11 | |
23 | 10 | adantr 274 | . . . . . . . . . . 11 |
24 | elnn 4514 | . . . . . . . . . . 11 | |
25 | 22, 23, 24 | syl2anc 408 | . . . . . . . . . 10 |
26 | 22 | iftrued 3476 | . . . . . . . . . . 11 |
27 | 3 | ffvelrnda 5548 | . . . . . . . . . . 11 |
28 | 26, 27 | eqeltrd 2214 | . . . . . . . . . 10 |
29 | 15, 21, 25, 28 | fvmptd3 5507 | . . . . . . . . 9 |
30 | 29, 26 | eqtrd 2170 | . . . . . . . 8 |
31 | 30 | eqeq2d 2149 | . . . . . . 7 |
32 | 31 | rexbidva 2432 | . . . . . 6 |
33 | 32 | adantr 274 | . . . . 5 |
34 | 18, 33 | mpbird 166 | . . . 4 |
35 | omelon 4517 | . . . . . . 7 | |
36 | 35 | onelssi 4346 | . . . . . 6 |
37 | ssrexv 3157 | . . . . . 6 | |
38 | 10, 36, 37 | 3syl 17 | . . . . 5 |
39 | 38 | adantr 274 | . . . 4 |
40 | 34, 39 | mpd 13 | . . 3 |
41 | 40 | ralrimiva 2503 | . 2 |
42 | dffo3 5560 | . 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 2414 wrex 2415 wss 3066 c0 3358 cif 3469 cmpt 3984 com 4499 wf 5114 wfo 5116 cfv 5118 |
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 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fo 5124 df-fv 5126 |
This theorem is referenced by: enumct 6993 |
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