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Mirrors > Home > NFE Home > Th. List > fnfreclem2 | Unicode version |
Description: Lemma for fnfrec 6320. Calculate the unique value of at zero. (Contributed by Scott Fenton, 31-Jul-2019.) |
Ref | Expression |
---|---|
fnfreclem2.1 | FRec |
fnfreclem2.2 | |
fnfreclem2.3 | |
fnfreclem2.4 |
Ref | Expression |
---|---|
fnfreclem2 | 0c |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 4640 | . 2 0c 0c | |
2 | snex 4111 | . . . 4 0c | |
3 | csucex 6259 | . . . . 5 1c | |
4 | fnfreclem2.2 | . . . . 5 | |
5 | pprodexg 5837 | . . . . 5 1c PProd 1c | |
6 | 3, 4, 5 | sylancr 644 | . . . 4 PProd 1c |
7 | fnfreclem2.1 | . . . . . 6 FRec | |
8 | df-frec 6310 | . . . . . 6 FRec Clos1 0c PProd 1c | |
9 | 7, 8 | eqtri 2373 | . . . . 5 Clos1 0c PProd 1c |
10 | 9 | clos1basesucg 5884 | . . . 4 0c PProd 1c 0c 0c 0c PProd 1c 0c |
11 | 2, 6, 10 | sylancr 644 | . . 3 0c 0c 0c PProd 1c 0c |
12 | 0cex 4392 | . . . . . . 7 0c | |
13 | fnfreclem2.3 | . . . . . . 7 | |
14 | opexg 4587 | . . . . . . 7 0c 0c | |
15 | 12, 13, 14 | sylancr 644 | . . . . . 6 0c |
16 | elsnc2g 3761 | . . . . . 6 0c 0c 0c 0c 0c | |
17 | 15, 16 | syl 15 | . . . . 5 0c 0c 0c 0c |
18 | opth 4602 | . . . . . 6 0c 0c 0c 0c | |
19 | 18 | simprbi 450 | . . . . 5 0c 0c |
20 | 17, 19 | syl6bi 219 | . . . 4 0c 0c |
21 | 0cnsuc 4401 | . . . . . . . . . . 11 Proj1 1c 0c | |
22 | df-ne 2518 | . . . . . . . . . . 11 Proj1 1c 0c Proj1 1c 0c | |
23 | 21, 22 | mpbi 199 | . . . . . . . . . 10 Proj1 1c 0c |
24 | 23 | intnanr 881 | . . . . . . . . 9 Proj1 1c 0c Proj2 |
25 | qrpprod 5836 | . . . . . . . . . 10 Proj1 Proj2 PProd 1c 0c Proj1 1c0c Proj2 | |
26 | opeq 4619 | . . . . . . . . . . 11 Proj1 Proj2 | |
27 | 26 | breq1i 4646 | . . . . . . . . . 10 PProd 1c 0c Proj1 Proj2 PProd 1c 0c |
28 | vex 2862 | . . . . . . . . . . . . . . 15 | |
29 | 28 | proj1ex 4593 | . . . . . . . . . . . . . 14 Proj1 |
30 | addceq1 4383 | . . . . . . . . . . . . . . 15 Proj1 1c Proj1 1c | |
31 | eqid 2353 | . . . . . . . . . . . . . . 15 1c 1c | |
32 | 1cex 4142 | . . . . . . . . . . . . . . . 16 1c | |
33 | 29, 32 | addcex 4394 | . . . . . . . . . . . . . . 15 Proj1 1c |
34 | 30, 31, 33 | fvmpt 5700 | . . . . . . . . . . . . . 14 Proj1 1c Proj1 Proj1 1c |
35 | 29, 34 | ax-mp 5 | . . . . . . . . . . . . 13 1c Proj1 Proj1 1c |
36 | 35 | eqeq1i 2360 | . . . . . . . . . . . 12 1c Proj1 0c Proj1 1c 0c |
37 | vex 2862 | . . . . . . . . . . . . . . 15 | |
38 | 37, 32 | addcex 4394 | . . . . . . . . . . . . . 14 1c |
39 | 38, 31 | fnmpti 5690 | . . . . . . . . . . . . 13 1c |
40 | fnbrfvb 5358 | . . . . . . . . . . . . 13 1c Proj1 1c Proj1 0c Proj1 1c0c | |
41 | 39, 29, 40 | mp2an 653 | . . . . . . . . . . . 12 1c Proj1 0c Proj1 1c0c |
42 | 36, 41 | bitr3i 242 | . . . . . . . . . . 11 Proj1 1c 0c Proj1 1c0c |
43 | 42 | anbi1i 676 | . . . . . . . . . 10 Proj1 1c 0c Proj2 Proj1 1c0c Proj2 |
44 | 25, 27, 43 | 3bitr4i 268 | . . . . . . . . 9 PProd 1c 0c Proj1 1c 0c Proj2 |
45 | 24, 44 | mtbir 290 | . . . . . . . 8 PProd 1c 0c |
46 | 45 | a1i 10 | . . . . . . 7 PProd 1c 0c |
47 | 46 | nrex 2716 | . . . . . 6 PProd 1c 0c |
48 | 47 | pm2.21i 123 | . . . . 5 PProd 1c 0c |
49 | 48 | a1i 10 | . . . 4 PProd 1c 0c |
50 | 20, 49 | jaod 369 | . . 3 0c 0c PProd 1c 0c |
51 | 11, 50 | sylbid 206 | . 2 0c |
52 | 1, 51 | syl5bi 208 | 1 0c |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 176 wo 357 wa 358 wceq 1642 wcel 1710 wne 2516 wrex 2615 cvv 2859 wss 3257 csn 3737 1cc1c 4134 0cc0c 4374 cplc 4375 cop 4561 Proj1 cproj1 4563 Proj2 cproj2 4564 class class class wbr 4639 cdm 4772 crn 4773 wfn 4776 cfv 4781 cmpt 5651 PProd cpprod 5737 Clos1 cclos1 5872 FRec cfrec 6309 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-csb 3137 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-iun 3971 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-swap 4724 df-sset 4725 df-co 4726 df-ima 4727 df-si 4728 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-fo 4793 df-fv 4795 df-2nd 4797 df-ov 5526 df-oprab 5528 df-mpt 5652 df-mpt2 5654 df-txp 5736 df-pprod 5738 df-fix 5740 df-cup 5742 df-disj 5744 df-addcfn 5746 df-ins2 5750 df-ins3 5752 df-image 5754 df-ins4 5756 df-si3 5758 df-clos1 5873 df-frec 6310 |
This theorem is referenced by: fnfrec 6320 |
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