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| Mirrors > Home > ILE Home > Th. List > apadd1 | Unicode version | ||
| Description: Addition respects apartness. Analogue of addcan 7942 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
| Ref | Expression |
|---|---|
| apadd1 |
     ![/\](wedge.gif "/\"){kind=small} 
   #
  #
|
are mutually distinct and distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cnre 7762 |
. . 3
     | |
| 2 | 1 | 3ad2ant3 1004 |
. 2
|
| 3 | cnre 7762 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1003 |
. . . . . 6
|
| 5 | 4 | ad2antrr 479 |
. . . . 5
|
| 6 | cnre 7762 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 1002 |
. . . . . . . . . 10
|
| 8 | 7 | ad2antrr 479 |
. . . . . . . . 9
|
| 9 | 8 | ad2antrr 479 |
. . . . . . . 8
|
| 10 | simplrl 524 |
. . . . . . . . . . . 12
| |
| 11 | simplrr 525 |
. . . . . . . . . . . 12
| |
| 12 | simprl 520 |
. . . . . . . . . . . . 13
| |
| 13 | 12 | ad3antrrr 483 |
. . . . . . . . . . . 12
|
| 14 | simprr 521 |
. . . . . . . . . . . . 13
| |
| 15 | 14 | ad3antrrr 483 |
. . . . . . . . . . . 12
|
| 16 | apreim 8365 |
. . . . . . . . . . . 12
#
# 
# | |
| 17 | 10, 11, 13, 15, 16 | syl22anc 1217 |
. . . . . . . . . . 11
#
#
# |
| 18 | simpr 109 |
. . . . . . . . . . . 12
| |
| 19 | simpllr 523 |
. . . . . . . . . . . 12
| |
| 20 | 18, 19 | breq12d 3942 |
. . . . . . . . . . 11
#
# |
| 21 | simprl 520 |
. . . . . . . . . . . . . . . 16
| |
| 22 | 21 | ad2antrr 479 |
. . . . . . . . . . . . . . 15
|
| 23 | 22 | ad3antrrr 483 |
. . . . . . . . . . . . . 14
|
| 24 | 10, 23 | readdcld 7795 |
. . . . . . . . . . . . 13
|
| 25 | simprr 521 |
. . . . . . . . . . . . . . . 16
| |
| 26 | 25 | ad2antrr 479 |
. . . . . . . . . . . . . . 15
|
| 27 | 26 | ad3antrrr 483 |
. . . . . . . . . . . . . 14
|
| 28 | 11, 27 | readdcld 7795 |
. . . . . . . . . . . . 13
|
| 29 | 13, 23 | readdcld 7795 |
. . . . . . . . . . . . 13
|
| 30 | 15, 27 | readdcld 7795 |
. . . . . . . . . . . . 13
|
| 31 | apreim 8365 |
. . . . . . . . . . . . 13
#
# # | |
| 32 | 24, 28, 29, 30, 31 | syl22anc 1217 |
. . . . . . . . . . . 12
#
# # |
| 33 | 10 | recnd 7794 |
. . . . . . . . . . . . . . 15
|
| 34 | ax-icn 7715 |
. . . . . . . . . . . . . . . . 17
| |
| 35 | 34 | a1i 9 |
. . . . . . . . . . . . . . . 16
|
| 36 | 11 | recnd 7794 |
. . . . . . . . . . . . . . . 16
|
| 37 | 35, 36 | mulcld 7786 |
. . . . . . . . . . . . . . 15
|
| 38 | 23 | recnd 7794 |
. . . . . . . . . . . . . . 15
|
| 39 | 27 | recnd 7794 |
. . . . . . . . . . . . . . . 16
|
| 40 | 35, 39 | mulcld 7786 |
. . . . . . . . . . . . . . 15
|
| 41 | 33, 37, 38, 40 | add4d 7931 |
. . . . . . . . . . . . . 14
|
| 42 | simplr 519 |
. . . . . . . . . . . . . . . 16
| |
| 43 | 42 | ad3antrrr 483 |
. . . . . . . . . . . . . . 15
|
| 44 | 18, 43 | oveq12d 5792 |
. . . . . . . . . . . . . 14
|
| 45 | 35, 36, 39 | adddid 7790 |
. . . . . . . . . . . . . . 15
|
| 46 | 45 | oveq2d 5790 |
. . . . . . . . . . . . . 14
|
| 47 | 41, 44, 46 | 3eqtr4d 2182 |
. . . . . . . . . . . . 13
|
| 48 | 13 | recnd 7794 |
. . . . . . . . . . . . . . 15
|
| 49 | 15 | recnd 7794 |
. . . . . . . . . . . . . . . 16
|
| 50 | 35, 49 | mulcld 7786 |
. . . . . . . . . . . . . . 15
|
| 51 | 48, 50, 38, 40 | add4d 7931 |
. . . . . . . . . . . . . 14
|
| 52 | 19, 43 | oveq12d 5792 |
. . . . . . . . . . . . . 14
|
| 53 | 35, 49, 39 | adddid 7790 |
. . . . . . . . . . . . . . 15
|
| 54 | 53 | oveq2d 5790 |
. . . . . . . . . . . . . 14
|
| 55 | 51, 52, 54 | 3eqtr4d 2182 |
. . . . . . . . . . . . 13
|
| 56 | 47, 55 | breq12d 3942 |
. . . . . . . . . . . 12
#
# |
| 57 | reapadd1 8358 |
. . . . . . . . . . . . . 14
#
# | |
| 58 | 10, 13, 23, 57 | syl3anc 1216 |
. . . . . . . . . . . . 13
#
# |
| 59 | reapadd1 8358 |
. . . . . . . . . . . . . 14
#
# | |
| 60 | 11, 15, 27, 59 | syl3anc 1216 |
. . . . . . . . . . . . 13
#
# |
| 61 | 58, 60 | orbi12d 782 |
. . . . . . . . . . . 12
# #
# # |
| 62 | 32, 56, 61 | 3bitr4d 219 |
. . . . . . . . . . 11
#
#
# |
| 63 | 17, 20, 62 | 3bitr4d 219 |
. . . . . . . . . 10
#
#
|
| 64 | 63 | ex 114 |
. . . . . . . . 9
#
#
|
| 65 | 64 | rexlimdvva 2557 |
. . . . . . . 8
#
#
|
| 66 | 9, 65 | mpd 13 |
. . . . . . 7
#
#
|
| 67 | 66 | ex 114 |
. . . . . 6
#
#
|
| 68 | 67 | rexlimdvva 2557 |
. . . . 5
#
#
|
| 69 | 5, 68 | mpd 13 |
. . . 4
#
#
|
| 70 | 69 | ex 114 |
. . 3
#
#
|
| 71 | 70 | rexlimdvva 2557 |
. 2
#
#
|
| 72 | 2, 71 | mpd 13 |
1
#
#
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 wo 697 w3a 962 wceq 1331 wcel 1480 wrex 2417 class class class wbr 3929
(class class class)co 5774 cc 7618 cr 7619 ci 7622 caddc 7623 cmul 7625 # cap 8343 |
| 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-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 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-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 |
| This theorem depends on definitions: df-bi 116 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-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 |
| This theorem is referenced by: apadd2 8371 addext 8372 apsub1 8404 subap0 8405 subap0d 8406 |
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