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Mirrors > Home > ILE Home > Th. List > enumctlemm | Unicode version |
Description: Lemma for enumct 7059. 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 5392 | . . . . . . 7 | |
3 | 1, 2 | syl 14 | . . . . . 6 |
4 | 3 | ffvelrnda 5602 | . . . . 5 |
5 | 4 | adantlr 469 | . . . 4 |
6 | enumctlemm.n0 | . . . . . 6 | |
7 | 3, 6 | ffvelrnd 5603 | . . . . 5 |
8 | 7 | ad2antrr 480 | . . . 4 |
9 | simpr 109 | . . . . 5 | |
10 | enumctlemm.n | . . . . . 6 | |
11 | 10 | adantr 274 | . . . . 5 |
12 | nndcel 6447 | . . . . 5 DECID | |
13 | 9, 11, 12 | syl2anc 409 | . . . 4 DECID |
14 | 5, 8, 13 | ifcldadc 3534 | . . 3 |
15 | enumctlemm.g | . . 3 | |
16 | 14, 15 | fmptd 5621 | . 2 |
17 | foelrn 5703 | . . . . . 6 | |
18 | 1, 17 | sylan 281 | . . . . 5 |
19 | eleq1w 2218 | . . . . . . . . . . 11 | |
20 | fveq2 5468 | . . . . . . . . . . 11 | |
21 | 19, 20 | ifbieq1d 3527 | . . . . . . . . . 10 |
22 | simpr 109 | . . . . . . . . . . 11 | |
23 | 10 | adantr 274 | . . . . . . . . . . 11 |
24 | elnn 4565 | . . . . . . . . . . 11 | |
25 | 22, 23, 24 | syl2anc 409 | . . . . . . . . . 10 |
26 | 22 | iftrued 3512 | . . . . . . . . . . 11 |
27 | 3 | ffvelrnda 5602 | . . . . . . . . . . 11 |
28 | 26, 27 | eqeltrd 2234 | . . . . . . . . . 10 |
29 | 15, 21, 25, 28 | fvmptd3 5561 | . . . . . . . . 9 |
30 | 29, 26 | eqtrd 2190 | . . . . . . . 8 |
31 | 30 | eqeq2d 2169 | . . . . . . 7 |
32 | 31 | rexbidva 2454 | . . . . . 6 |
33 | 32 | adantr 274 | . . . . 5 |
34 | 18, 33 | mpbird 166 | . . . 4 |
35 | omelon 4568 | . . . . . . 7 | |
36 | 35 | onelssi 4389 | . . . . . 6 |
37 | ssrexv 3193 | . . . . . 6 | |
38 | 10, 36, 37 | 3syl 17 | . . . . 5 |
39 | 38 | adantr 274 | . . . 4 |
40 | 34, 39 | mpd 13 | . . 3 |
41 | 40 | ralrimiva 2530 | . 2 |
42 | dffo3 5614 | . 2 | |
43 | 16, 41, 42 | sylanbrc 414 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 DECID wdc 820 wceq 1335 wcel 2128 wral 2435 wrex 2436 wss 3102 c0 3394 cif 3505 cmpt 4025 com 4549 wf 5166 wfo 5168 cfv 5170 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-fo 5176 df-fv 5178 |
This theorem is referenced by: enumct 7059 |
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