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Mirrors > Home > ILE Home > Th. List > enumctlemm | Unicode version |
Description: Lemma for enumct 6968. 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 5315 | . . . . . . 7 | |
3 | 1, 2 | syl 14 | . . . . . 6 |
4 | 3 | ffvelrnda 5523 | . . . . 5 |
5 | 4 | adantlr 468 | . . . 4 |
6 | enumctlemm.n0 | . . . . . 6 | |
7 | 3, 6 | ffvelrnd 5524 | . . . . 5 |
8 | 7 | ad2antrr 479 | . . . 4 |
9 | simpr 109 | . . . . 5 | |
10 | enumctlemm.n | . . . . . 6 | |
11 | 10 | adantr 274 | . . . . 5 |
12 | nndcel 6364 | . . . . 5 DECID | |
13 | 9, 11, 12 | syl2anc 408 | . . . 4 DECID |
14 | 5, 8, 13 | ifcldadc 3471 | . . 3 |
15 | enumctlemm.g | . . 3 | |
16 | 14, 15 | fmptd 5542 | . 2 |
17 | foelrn 5622 | . . . . . 6 | |
18 | 1, 17 | sylan 281 | . . . . 5 |
19 | eleq1w 2178 | . . . . . . . . . . 11 | |
20 | fveq2 5389 | . . . . . . . . . . 11 | |
21 | 19, 20 | ifbieq1d 3464 | . . . . . . . . . 10 |
22 | simpr 109 | . . . . . . . . . . 11 | |
23 | 10 | adantr 274 | . . . . . . . . . . 11 |
24 | elnn 4489 | . . . . . . . . . . 11 | |
25 | 22, 23, 24 | syl2anc 408 | . . . . . . . . . 10 |
26 | 22 | iftrued 3451 | . . . . . . . . . . 11 |
27 | 3 | ffvelrnda 5523 | . . . . . . . . . . 11 |
28 | 26, 27 | eqeltrd 2194 | . . . . . . . . . 10 |
29 | 15, 21, 25, 28 | fvmptd3 5482 | . . . . . . . . 9 |
30 | 29, 26 | eqtrd 2150 | . . . . . . . 8 |
31 | 30 | eqeq2d 2129 | . . . . . . 7 |
32 | 31 | rexbidva 2411 | . . . . . 6 |
33 | 32 | adantr 274 | . . . . 5 |
34 | 18, 33 | mpbird 166 | . . . 4 |
35 | omelon 4492 | . . . . . . 7 | |
36 | 35 | onelssi 4321 | . . . . . 6 |
37 | ssrexv 3132 | . . . . . 6 | |
38 | 10, 36, 37 | 3syl 17 | . . . . 5 |
39 | 38 | adantr 274 | . . . 4 |
40 | 34, 39 | mpd 13 | . . 3 |
41 | 40 | ralrimiva 2482 | . 2 |
42 | dffo3 5535 | . 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 804 wceq 1316 wcel 1465 wral 2393 wrex 2394 wss 3041 c0 3333 cif 3444 cmpt 3959 com 4474 wf 5089 wfo 5091 cfv 5093 |
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-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 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-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 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-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 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-fo 5099 df-fv 5101 |
This theorem is referenced by: enumct 6968 |
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