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Mirrors > Home > ILE Home > Th. List > frec2uzsucd | Unicode version |
Description: The value of (see frec2uz0d 10172) at a successor. (Contributed by Jim Kingdon, 16-May-2020.) |
Ref | Expression |
---|---|
frec2uz.1 | |
frec2uz.2 | frec |
frec2uzzd.a |
Ref | Expression |
---|---|
frec2uzsucd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2z 9090 | . . . . . . 7 | |
2 | oveq1 5781 | . . . . . . . 8 | |
3 | eqid 2139 | . . . . . . . 8 | |
4 | 2, 3 | fvmptg 5497 | . . . . . . 7 |
5 | 1, 4 | mpdan 417 | . . . . . 6 |
6 | 5, 1 | eqeltrd 2216 | . . . . 5 |
7 | 6 | rgen 2485 | . . . 4 |
8 | frec2uz.1 | . . . 4 | |
9 | frec2uzzd.a | . . . 4 | |
10 | frecsuc 6304 | . . . 4 frec frec | |
11 | 7, 8, 9, 10 | mp3an2i 1320 | . . 3 frec frec |
12 | frec2uz.2 | . . . 4 frec | |
13 | 12 | fveq1i 5422 | . . 3 frec |
14 | 12 | fveq1i 5422 | . . . 4 frec |
15 | 14 | fveq2i 5424 | . . 3 frec |
16 | 11, 13, 15 | 3eqtr4g 2197 | . 2 |
17 | 8, 12, 9 | frec2uzzd 10173 | . . 3 |
18 | oveq1 5781 | . . . 4 | |
19 | 2 | cbvmptv 4024 | . . . 4 |
20 | 18, 19, 1 | fvmpt3 5500 | . . 3 |
21 | 17, 20 | syl 14 | . 2 |
22 | 16, 21 | eqtrd 2172 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 wcel 1480 wral 2416 cmpt 3989 csuc 4287 com 4504 cfv 5123 (class class class)co 5774 freccfrec 6287 c1 7621 caddc 7623 cz 9054 |
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 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-recs 6202 df-frec 6288 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 |
This theorem is referenced by: frec2uzuzd 10175 frec2uzltd 10176 frec2uzrand 10178 frec2uzrdg 10182 frecuzrdgsuc 10187 frecuzrdgg 10189 frecfzennn 10199 1tonninf 10213 omgadd 10548 ennnfonelemkh 11925 ennnfonelemhf1o 11926 ennnfonelemnn0 11935 isomninnlem 13225 |
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