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| Mirrors > Home > ILE Home > Th. List > apadd1 | Unicode version | ||
| Description: Addition respects apartness. Analogue of addcan 8131 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 7948 |
. . 3
     | |
| 2 | 1 | 3ad2ant3 1020 |
. 2
|
| 3 | cnre 7948 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1019 |
. . . . . 6
|
| 5 | 4 | ad2antrr 488 |
. . . . 5
|
| 6 | cnre 7948 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 1018 |
. . . . . . . . . 10
|
| 8 | 7 | ad2antrr 488 |
. . . . . . . . 9
|
| 9 | 8 | ad2antrr 488 |
. . . . . . . 8
|
| 10 | simplrl 535 |
. . . . . . . . . . . 12
| |
| 11 | simplrr 536 |
. . . . . . . . . . . 12
| |
| 12 | simprl 529 |
. . . . . . . . . . . . 13
| |
| 13 | 12 | ad3antrrr 492 |
. . . . . . . . . . . 12
|
| 14 | simprr 531 |
. . . . . . . . . . . . 13
| |
| 15 | 14 | ad3antrrr 492 |
. . . . . . . . . . . 12
|
| 16 | apreim 8554 |
. . . . . . . . . . . 12
#
# 
# | |
| 17 | 10, 11, 13, 15, 16 | syl22anc 1239 |
. . . . . . . . . . 11
#
#
# |
| 18 | simpr 110 |
. . . . . . . . . . . 12
| |
| 19 | simpllr 534 |
. . . . . . . . . . . 12
| |
| 20 | 18, 19 | breq12d 4014 |
. . . . . . . . . . 11
#
# |
| 21 | simprl 529 |
. . . . . . . . . . . . . . . 16
| |
| 22 | 21 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
|
| 23 | 22 | ad3antrrr 492 |
. . . . . . . . . . . . . 14
|
| 24 | 10, 23 | readdcld 7981 |
. . . . . . . . . . . . 13
|
| 25 | simprr 531 |
. . . . . . . . . . . . . . . 16
| |
| 26 | 25 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
|
| 27 | 26 | ad3antrrr 492 |
. . . . . . . . . . . . . 14
|
| 28 | 11, 27 | readdcld 7981 |
. . . . . . . . . . . . 13
|
| 29 | 13, 23 | readdcld 7981 |
. . . . . . . . . . . . 13
|
| 30 | 15, 27 | readdcld 7981 |
. . . . . . . . . . . . 13
|
| 31 | apreim 8554 |
. . . . . . . . . . . . 13
#
# # | |
| 32 | 24, 28, 29, 30, 31 | syl22anc 1239 |
. . . . . . . . . . . 12
#
# # |
| 33 | 10 | recnd 7980 |
. . . . . . . . . . . . . . 15
|
| 34 | ax-icn 7901 |
. . . . . . . . . . . . . . . . 17
| |
| 35 | 34 | a1i 9 |
. . . . . . . . . . . . . . . 16
|
| 36 | 11 | recnd 7980 |
. . . . . . . . . . . . . . . 16
|
| 37 | 35, 36 | mulcld 7972 |
. . . . . . . . . . . . . . 15
|
| 38 | 23 | recnd 7980 |
. . . . . . . . . . . . . . 15
|
| 39 | 27 | recnd 7980 |
. . . . . . . . . . . . . . . 16
|
| 40 | 35, 39 | mulcld 7972 |
. . . . . . . . . . . . . . 15
|
| 41 | 33, 37, 38, 40 | add4d 8120 |
. . . . . . . . . . . . . 14
|
| 42 | simplr 528 |
. . . . . . . . . . . . . . . 16
| |
| 43 | 42 | ad3antrrr 492 |
. . . . . . . . . . . . . . 15
|
| 44 | 18, 43 | oveq12d 5888 |
. . . . . . . . . . . . . 14
|
| 45 | 35, 36, 39 | adddid 7976 |
. . . . . . . . . . . . . . 15
|
| 46 | 45 | oveq2d 5886 |
. . . . . . . . . . . . . 14
|
| 47 | 41, 44, 46 | 3eqtr4d 2220 |
. . . . . . . . . . . . 13
|
| 48 | 13 | recnd 7980 |
. . . . . . . . . . . . . . 15
|
| 49 | 15 | recnd 7980 |
. . . . . . . . . . . . . . . 16
|
| 50 | 35, 49 | mulcld 7972 |
. . . . . . . . . . . . . . 15
|
| 51 | 48, 50, 38, 40 | add4d 8120 |
. . . . . . . . . . . . . 14
|
| 52 | 19, 43 | oveq12d 5888 |
. . . . . . . . . . . . . 14
|
| 53 | 35, 49, 39 | adddid 7976 |
. . . . . . . . . . . . . . 15
|
| 54 | 53 | oveq2d 5886 |
. . . . . . . . . . . . . 14
|
| 55 | 51, 52, 54 | 3eqtr4d 2220 |
. . . . . . . . . . . . 13
|
| 56 | 47, 55 | breq12d 4014 |
. . . . . . . . . . . 12
#
# |
| 57 | reapadd1 8547 |
. . . . . . . . . . . . . 14
#
# | |
| 58 | 10, 13, 23, 57 | syl3anc 1238 |
. . . . . . . . . . . . 13
#
# |
| 59 | reapadd1 8547 |
. . . . . . . . . . . . . 14
#
# | |
| 60 | 11, 15, 27, 59 | syl3anc 1238 |
. . . . . . . . . . . . 13
#
# |
| 61 | 58, 60 | orbi12d 793 |
. . . . . . . . . . . 12
# #
# # |
| 62 | 32, 56, 61 | 3bitr4d 220 |
. . . . . . . . . . 11
#
#
# |
| 63 | 17, 20, 62 | 3bitr4d 220 |
. . . . . . . . . 10
#
#
|
| 64 | 63 | ex 115 |
. . . . . . . . 9
#
#
|
| 65 | 64 | rexlimdvva 2602 |
. . . . . . . 8
#
#
|
| 66 | 9, 65 | mpd 13 |
. . . . . . 7
#
#
|
| 67 | 66 | ex 115 |
. . . . . 6
#
#
|
| 68 | 67 | rexlimdvva 2602 |
. . . . 5
#
#
|
| 69 | 5, 68 | mpd 13 |
. . . 4
#
#
|
| 70 | 69 | ex 115 |
. . 3
#
#
|
| 71 | 70 | rexlimdvva 2602 |
. 2
#
#
|
| 72 | 2, 71 | mpd 13 |
1
#
#
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wo 708 w3a 978 wceq 1353 wcel 2148 wrex 2456 class class class wbr 4001
(class class class)co 5870 cc 7804 cr 7805 ci 7808 caddc 7809 cmul 7811 # cap 8532 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4119 ax-pow 4172 ax-pr 4207 ax-un 4431 ax-setind 4534 ax-cnex 7897 ax-resscn 7898 ax-1cn 7899 ax-1re 7900 ax-icn 7901 ax-addcl 7902 ax-addrcl 7903 ax-mulcl 7904 ax-mulrcl 7905 ax-addcom 7906 ax-mulcom 7907 ax-addass 7908 ax-mulass 7909 ax-distr 7910 ax-i2m1 7911 ax-0lt1 7912 ax-1rid 7913 ax-0id 7914 ax-rnegex 7915 ax-precex 7916 ax-cnre 7917 ax-pre-ltirr 7918 ax-pre-lttrn 7920 ax-pre-apti 7921 ax-pre-ltadd 7922 ax-pre-mulgt0 7923 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3809 df-br 4002 df-opab 4063 df-id 4291 df-xp 4630 df-rel 4631 df-cnv 4632 df-co 4633 df-dm 4634 df-iota 5175 df-fun 5215 df-fv 5221 df-riota 5826 df-ov 5873 df-oprab 5874 df-mpo 5875 df-pnf 7988 df-mnf 7989 df-ltxr 7991 df-sub 8124 df-neg 8125 df-reap 8526 df-ap 8533 |
| This theorem is referenced by: apadd2 8560 addext 8561 apsub1 8593 subap0 8594 |
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