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| Mirrors > Home > ILE Home > Th. List > apadd1 | Unicode version | ||
| Description: Addition respects apartness. Analogue of addcan 8078 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 7895 |
. . 3
     | |
| 2 | 1 | 3ad2ant3 1010 |
. 2
|
| 3 | cnre 7895 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1009 |
. . . . . 6
|
| 5 | 4 | ad2antrr 480 |
. . . . 5
|
| 6 | cnre 7895 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 1008 |
. . . . . . . . . 10
|
| 8 | 7 | ad2antrr 480 |
. . . . . . . . 9
|
| 9 | 8 | ad2antrr 480 |
. . . . . . . 8
|
| 10 | simplrl 525 |
. . . . . . . . . . . 12
| |
| 11 | simplrr 526 |
. . . . . . . . . . . 12
| |
| 12 | simprl 521 |
. . . . . . . . . . . . 13
| |
| 13 | 12 | ad3antrrr 484 |
. . . . . . . . . . . 12
|
| 14 | simprr 522 |
. . . . . . . . . . . . 13
| |
| 15 | 14 | ad3antrrr 484 |
. . . . . . . . . . . 12
|
| 16 | apreim 8501 |
. . . . . . . . . . . 12
#
# 
# | |
| 17 | 10, 11, 13, 15, 16 | syl22anc 1229 |
. . . . . . . . . . 11
#
#
# |
| 18 | simpr 109 |
. . . . . . . . . . . 12
| |
| 19 | simpllr 524 |
. . . . . . . . . . . 12
| |
| 20 | 18, 19 | breq12d 3995 |
. . . . . . . . . . 11
#
# |
| 21 | simprl 521 |
. . . . . . . . . . . . . . . 16
| |
| 22 | 21 | ad2antrr 480 |
. . . . . . . . . . . . . . 15
|
| 23 | 22 | ad3antrrr 484 |
. . . . . . . . . . . . . 14
|
| 24 | 10, 23 | readdcld 7928 |
. . . . . . . . . . . . 13
|
| 25 | simprr 522 |
. . . . . . . . . . . . . . . 16
| |
| 26 | 25 | ad2antrr 480 |
. . . . . . . . . . . . . . 15
|
| 27 | 26 | ad3antrrr 484 |
. . . . . . . . . . . . . 14
|
| 28 | 11, 27 | readdcld 7928 |
. . . . . . . . . . . . 13
|
| 29 | 13, 23 | readdcld 7928 |
. . . . . . . . . . . . 13
|
| 30 | 15, 27 | readdcld 7928 |
. . . . . . . . . . . . 13
|
| 31 | apreim 8501 |
. . . . . . . . . . . . 13
#
# # | |
| 32 | 24, 28, 29, 30, 31 | syl22anc 1229 |
. . . . . . . . . . . 12
#
# # |
| 33 | 10 | recnd 7927 |
. . . . . . . . . . . . . . 15
|
| 34 | ax-icn 7848 |
. . . . . . . . . . . . . . . . 17
| |
| 35 | 34 | a1i 9 |
. . . . . . . . . . . . . . . 16
|
| 36 | 11 | recnd 7927 |
. . . . . . . . . . . . . . . 16
|
| 37 | 35, 36 | mulcld 7919 |
. . . . . . . . . . . . . . 15
|
| 38 | 23 | recnd 7927 |
. . . . . . . . . . . . . . 15
|
| 39 | 27 | recnd 7927 |
. . . . . . . . . . . . . . . 16
|
| 40 | 35, 39 | mulcld 7919 |
. . . . . . . . . . . . . . 15
|
| 41 | 33, 37, 38, 40 | add4d 8067 |
. . . . . . . . . . . . . 14
|
| 42 | simplr 520 |
. . . . . . . . . . . . . . . 16
| |
| 43 | 42 | ad3antrrr 484 |
. . . . . . . . . . . . . . 15
|
| 44 | 18, 43 | oveq12d 5860 |
. . . . . . . . . . . . . 14
|
| 45 | 35, 36, 39 | adddid 7923 |
. . . . . . . . . . . . . . 15
|
| 46 | 45 | oveq2d 5858 |
. . . . . . . . . . . . . 14
|
| 47 | 41, 44, 46 | 3eqtr4d 2208 |
. . . . . . . . . . . . 13
|
| 48 | 13 | recnd 7927 |
. . . . . . . . . . . . . . 15
|
| 49 | 15 | recnd 7927 |
. . . . . . . . . . . . . . . 16
|
| 50 | 35, 49 | mulcld 7919 |
. . . . . . . . . . . . . . 15
|
| 51 | 48, 50, 38, 40 | add4d 8067 |
. . . . . . . . . . . . . 14
|
| 52 | 19, 43 | oveq12d 5860 |
. . . . . . . . . . . . . 14
|
| 53 | 35, 49, 39 | adddid 7923 |
. . . . . . . . . . . . . . 15
|
| 54 | 53 | oveq2d 5858 |
. . . . . . . . . . . . . 14
|
| 55 | 51, 52, 54 | 3eqtr4d 2208 |
. . . . . . . . . . . . 13
|
| 56 | 47, 55 | breq12d 3995 |
. . . . . . . . . . . 12
#
# |
| 57 | reapadd1 8494 |
. . . . . . . . . . . . . 14
#
# | |
| 58 | 10, 13, 23, 57 | syl3anc 1228 |
. . . . . . . . . . . . 13
#
# |
| 59 | reapadd1 8494 |
. . . . . . . . . . . . . 14
#
# | |
| 60 | 11, 15, 27, 59 | syl3anc 1228 |
. . . . . . . . . . . . 13
#
# |
| 61 | 58, 60 | orbi12d 783 |
. . . . . . . . . . . 12
# #
# # |
| 62 | 32, 56, 61 | 3bitr4d 219 |
. . . . . . . . . . 11
#
#
# |
| 63 | 17, 20, 62 | 3bitr4d 219 |
. . . . . . . . . 10
#
#
|
| 64 | 63 | ex 114 |
. . . . . . . . 9
#
#
|
| 65 | 64 | rexlimdvva 2591 |
. . . . . . . 8
#
#
|
| 66 | 9, 65 | mpd 13 |
. . . . . . 7
#
#
|
| 67 | 66 | ex 114 |
. . . . . 6
#
#
|
| 68 | 67 | rexlimdvva 2591 |
. . . . 5
#
#
|
| 69 | 5, 68 | mpd 13 |
. . . 4
#
#
|
| 70 | 69 | ex 114 |
. . 3
#
#
|
| 71 | 70 | rexlimdvva 2591 |
. 2
#
#
|
| 72 | 2, 71 | mpd 13 |
1
#
#
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 wo 698 w3a 968 wceq 1343 wcel 2136 wrex 2445 class class class wbr 3982
(class class class)co 5842 cc 7751 cr 7752 ci 7755 caddc 7756 cmul 7758 # cap 8479 |
| 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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 |
| This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-iota 5153 df-fun 5190 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-pnf 7935 df-mnf 7936 df-ltxr 7938 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 |
| This theorem is referenced by: apadd2 8507 addext 8508 apsub1 8540 subap0 8541 |
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