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Mirrors > Home > ILE Home > Th. List > enumctlemm | Unicode version |
Description: Lemma for enumct 7092. 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 5420 | . . . . . . 7 | |
3 | 1, 2 | syl 14 | . . . . . 6 |
4 | 3 | ffvelrnda 5631 | . . . . 5 |
5 | 4 | adantlr 474 | . . . 4 |
6 | enumctlemm.n0 | . . . . . 6 | |
7 | 3, 6 | ffvelrnd 5632 | . . . . 5 |
8 | 7 | ad2antrr 485 | . . . 4 |
9 | simpr 109 | . . . . 5 | |
10 | enumctlemm.n | . . . . . 6 | |
11 | 10 | adantr 274 | . . . . 5 |
12 | nndcel 6479 | . . . . 5 DECID | |
13 | 9, 11, 12 | syl2anc 409 | . . . 4 DECID |
14 | 5, 8, 13 | ifcldadc 3555 | . . 3 |
15 | enumctlemm.g | . . 3 | |
16 | 14, 15 | fmptd 5650 | . 2 |
17 | foelrn 5732 | . . . . . 6 | |
18 | 1, 17 | sylan 281 | . . . . 5 |
19 | eleq1w 2231 | . . . . . . . . . . 11 | |
20 | fveq2 5496 | . . . . . . . . . . 11 | |
21 | 19, 20 | ifbieq1d 3548 | . . . . . . . . . 10 |
22 | simpr 109 | . . . . . . . . . . 11 | |
23 | 10 | adantr 274 | . . . . . . . . . . 11 |
24 | elnn 4590 | . . . . . . . . . . 11 | |
25 | 22, 23, 24 | syl2anc 409 | . . . . . . . . . 10 |
26 | 22 | iftrued 3533 | . . . . . . . . . . 11 |
27 | 3 | ffvelrnda 5631 | . . . . . . . . . . 11 |
28 | 26, 27 | eqeltrd 2247 | . . . . . . . . . 10 |
29 | 15, 21, 25, 28 | fvmptd3 5589 | . . . . . . . . 9 |
30 | 29, 26 | eqtrd 2203 | . . . . . . . 8 |
31 | 30 | eqeq2d 2182 | . . . . . . 7 |
32 | 31 | rexbidva 2467 | . . . . . 6 |
33 | 32 | adantr 274 | . . . . 5 |
34 | 18, 33 | mpbird 166 | . . . 4 |
35 | omelon 4593 | . . . . . . 7 | |
36 | 35 | onelssi 4414 | . . . . . 6 |
37 | ssrexv 3212 | . . . . . 6 | |
38 | 10, 36, 37 | 3syl 17 | . . . . 5 |
39 | 38 | adantr 274 | . . . 4 |
40 | 34, 39 | mpd 13 | . . 3 |
41 | 40 | ralrimiva 2543 | . 2 |
42 | dffo3 5643 | . 2 | |
43 | 16, 41, 42 | sylanbrc 415 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 DECID wdc 829 wceq 1348 wcel 2141 wral 2448 wrex 2449 wss 3121 c0 3414 cif 3526 cmpt 4050 com 4574 wf 5194 wfo 5196 cfv 5198 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fo 5204 df-fv 5206 |
This theorem is referenced by: enumct 7092 |
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