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| Mirrors > Home > ILE Home > Th. List > apadd1 | Unicode version | ||
| Description: Addition respects apartness. Analogue of addcan 8506 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 8322 |
. . 3
     | |
| 2 | 1 | 3ad2ant3 1051 |
. 2
|
| 3 | cnre 8322 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1050 |
. . . . . 6
|
| 5 | 4 | ad2antrr 492 |
. . . . 5
|
| 6 | cnre 8322 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 1049 |
. . . . . . . . . 10
|
| 8 | 7 | ad2antrr 492 |
. . . . . . . . 9
|
| 9 | 8 | ad2antrr 492 |
. . . . . . . 8
|
| 10 | simplrl 541 |
. . . . . . . . . . . 12
| |
| 11 | simplrr 542 |
. . . . . . . . . . . 12
| |
| 12 | simprl 535 |
. . . . . . . . . . . . 13
| |
| 13 | 12 | ad3antrrr 496 |
. . . . . . . . . . . 12
|
| 14 | simprr 537 |
. . . . . . . . . . . . 13
| |
| 15 | 14 | ad3antrrr 496 |
. . . . . . . . . . . 12
|
| 16 | apreim 8931 |
. . . . . . . . . . . 12
#
# 
# | |
| 17 | 10, 11, 13, 15, 16 | syl22anc 1279 |
. . . . . . . . . . 11
#
#
# |
| 18 | simpr 110 |
. . . . . . . . . . . 12
| |
| 19 | simpllr 540 |
. . . . . . . . . . . 12
| |
| 20 | 18, 19 | breq12d 4143 |
. . . . . . . . . . 11
#
# |
| 21 | simprl 535 |
. . . . . . . . . . . . . . . 16
| |
| 22 | 21 | ad2antrr 492 |
. . . . . . . . . . . . . . 15
|
| 23 | 22 | ad3antrrr 496 |
. . . . . . . . . . . . . 14
|
| 24 | 10, 23 | readdcld 8355 |
. . . . . . . . . . . . 13
|
| 25 | simprr 537 |
. . . . . . . . . . . . . . . 16
| |
| 26 | 25 | ad2antrr 492 |
. . . . . . . . . . . . . . 15
|
| 27 | 26 | ad3antrrr 496 |
. . . . . . . . . . . . . 14
|
| 28 | 11, 27 | readdcld 8355 |
. . . . . . . . . . . . 13
|
| 29 | 13, 23 | readdcld 8355 |
. . . . . . . . . . . . 13
|
| 30 | 15, 27 | readdcld 8355 |
. . . . . . . . . . . . 13
|
| 31 | apreim 8931 |
. . . . . . . . . . . . 13
#
# # | |
| 32 | 24, 28, 29, 30, 31 | syl22anc 1279 |
. . . . . . . . . . . 12
#
# # |
| 33 | 10 | recnd 8354 |
. . . . . . . . . . . . . . 15
|
| 34 | ax-icn 8274 |
. . . . . . . . . . . . . . . . 17
| |
| 35 | 34 | a1i 9 |
. . . . . . . . . . . . . . . 16
|
| 36 | 11 | recnd 8354 |
. . . . . . . . . . . . . . . 16
|
| 37 | 35, 36 | mulcld 8346 |
. . . . . . . . . . . . . . 15
|
| 38 | 23 | recnd 8354 |
. . . . . . . . . . . . . . 15
|
| 39 | 27 | recnd 8354 |
. . . . . . . . . . . . . . . 16
|
| 40 | 35, 39 | mulcld 8346 |
. . . . . . . . . . . . . . 15
|
| 41 | 33, 37, 38, 40 | add4d 8495 |
. . . . . . . . . . . . . 14
|
| 42 | simplr 533 |
. . . . . . . . . . . . . . . 16
| |
| 43 | 42 | ad3antrrr 496 |
. . . . . . . . . . . . . . 15
|
| 44 | 18, 43 | oveq12d 6103 |
. . . . . . . . . . . . . 14
|
| 45 | 35, 36, 39 | adddid 8350 |
. . . . . . . . . . . . . . 15
|
| 46 | 45 | oveq2d 6101 |
. . . . . . . . . . . . . 14
|
| 47 | 41, 44, 46 | 3eqtr4d 2281 |
. . . . . . . . . . . . 13
|
| 48 | 13 | recnd 8354 |
. . . . . . . . . . . . . . 15
|
| 49 | 15 | recnd 8354 |
. . . . . . . . . . . . . . . 16
|
| 50 | 35, 49 | mulcld 8346 |
. . . . . . . . . . . . . . 15
|
| 51 | 48, 50, 38, 40 | add4d 8495 |
. . . . . . . . . . . . . 14
|
| 52 | 19, 43 | oveq12d 6103 |
. . . . . . . . . . . . . 14
|
| 53 | 35, 49, 39 | adddid 8350 |
. . . . . . . . . . . . . . 15
|
| 54 | 53 | oveq2d 6101 |
. . . . . . . . . . . . . 14
|
| 55 | 51, 52, 54 | 3eqtr4d 2281 |
. . . . . . . . . . . . 13
|
| 56 | 47, 55 | breq12d 4143 |
. . . . . . . . . . . 12
#
# |
| 57 | reapadd1 8924 |
. . . . . . . . . . . . . 14
#
# | |
| 58 | 10, 13, 23, 57 | syl3anc 1278 |
. . . . . . . . . . . . 13
#
# |
| 59 | reapadd1 8924 |
. . . . . . . . . . . . . 14
#
# | |
| 60 | 11, 15, 27, 59 | syl3anc 1278 |
. . . . . . . . . . . . 13
#
# |
| 61 | 58, 60 | orbi12d 805 |
. . . . . . . . . . . 12
# #
# # |
| 62 | 32, 56, 61 | 3bitr4d 220 |
. . . . . . . . . . 11
#
#
# |
| 63 | 17, 20, 62 | 3bitr4d 220 |
. . . . . . . . . 10
#
#
|
| 64 | 63 | ex 115 |
. . . . . . . . 9
#
#
|
| 65 | 64 | rexlimdvva 2676 |
. . . . . . . 8
#
#
|
| 66 | 9, 65 | mpd 13 |
. . . . . . 7
#
#
|
| 67 | 66 | ex 115 |
. . . . . 6
#
#
|
| 68 | 67 | rexlimdvva 2676 |
. . . . 5
#
#
|
| 69 | 5, 68 | mpd 13 |
. . . 4
#
#
|
| 70 | 69 | ex 115 |
. . 3
#
#
|
| 71 | 70 | rexlimdvva 2676 |
. 2
#
#
|
| 72 | 2, 71 | mpd 13 |
1
#
#
|
| Colors of variables: wff set class |
| This proof depends on syntax axioms:
wi 4
wa 104
wb 105 wo 720 w3a 1009 wceq 1402 wcel 2209 wrex 2529 class class class wbr 4130
(class class class)co 6085 cc 8177 cr 8178 ci 8181 caddc 8182 cmul 8184 # cap 8909 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 |
| This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4131 df-opab 4193 df-id 4438 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-iota 5337 df-fun 5379 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-pnf 8362 df-mnf 8363 df-ltxr 8365 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 |
| This theorem is used by: apadd2 8937 addext 8938 apsub1 8970 subap0 8971 |
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