Nearby theorems
|
|
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > ILE Home > Th. List > apadd1 | Unicode version | ||
| Description: Addition respects apartness. Analogue of addcan 7288 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 7115 |
. . 3
     | |
| 2 | 1 | 3ad2ant3 961 |
. 2
|
| 3 | cnre 7115 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 960 |
. . . . . 6
|
| 5 | 4 | ad2antrr 471 |
. . . . 5
|
| 6 | cnre 7115 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 959 |
. . . . . . . . . 10
|
| 8 | 7 | ad2antrr 471 |
. . . . . . . . 9
|
| 9 | 8 | ad2antrr 471 |
. . . . . . . 8
|
| 10 | simplrl 501 |
. . . . . . . . . . . 12
| |
| 11 | simplrr 502 |
. . . . . . . . . . . 12
| |
| 12 | simprl 497 |
. . . . . . . . . . . . 13
| |
| 13 | 12 | ad3antrrr 475 |
. . . . . . . . . . . 12
|
| 14 | simprr 498 |
. . . . . . . . . . . . 13
| |
| 15 | 14 | ad3antrrr 475 |
. . . . . . . . . . . 12
|
| 16 | apreim 7703 |
. . . . . . . . . . . 12
#
# 
# | |
| 17 | 10, 11, 13, 15, 16 | syl22anc 1170 |
. . . . . . . . . . 11
#
#
# |
| 18 | simpr 108 |
. . . . . . . . . . . 12
| |
| 19 | simpllr 500 |
. . . . . . . . . . . 12
| |
| 20 | 18, 19 | breq12d 3798 |
. . . . . . . . . . 11
#
# |
| 21 | simprl 497 |
. . . . . . . . . . . . . . . 16
| |
| 22 | 21 | ad2antrr 471 |
. . . . . . . . . . . . . . 15
|
| 23 | 22 | ad3antrrr 475 |
. . . . . . . . . . . . . 14
|
| 24 | 10, 23 | readdcld 7148 |
. . . . . . . . . . . . 13
|
| 25 | simprr 498 |
. . . . . . . . . . . . . . . 16
| |
| 26 | 25 | ad2antrr 471 |
. . . . . . . . . . . . . . 15
|
| 27 | 26 | ad3antrrr 475 |
. . . . . . . . . . . . . 14
|
| 28 | 11, 27 | readdcld 7148 |
. . . . . . . . . . . . 13
|
| 29 | 13, 23 | readdcld 7148 |
. . . . . . . . . . . . 13
|
| 30 | 15, 27 | readdcld 7148 |
. . . . . . . . . . . . 13
|
| 31 | apreim 7703 |
. . . . . . . . . . . . 13
#
# # | |
| 32 | 24, 28, 29, 30, 31 | syl22anc 1170 |
. . . . . . . . . . . 12
#
# # |
| 33 | 10 | recnd 7147 |
. . . . . . . . . . . . . . 15
|
| 34 | ax-icn 7071 |
. . . . . . . . . . . . . . . . 17
| |
| 35 | 34 | a1i 9 |
. . . . . . . . . . . . . . . 16
|
| 36 | 11 | recnd 7147 |
. . . . . . . . . . . . . . . 16
|
| 37 | 35, 36 | mulcld 7139 |
. . . . . . . . . . . . . . 15
|
| 38 | 23 | recnd 7147 |
. . . . . . . . . . . . . . 15
|
| 39 | 27 | recnd 7147 |
. . . . . . . . . . . . . . . 16
|
| 40 | 35, 39 | mulcld 7139 |
. . . . . . . . . . . . . . 15
|
| 41 | 33, 37, 38, 40 | add4d 7277 |
. . . . . . . . . . . . . 14
|
| 42 | simplr 496 |
. . . . . . . . . . . . . . . 16
| |
| 43 | 42 | ad3antrrr 475 |
. . . . . . . . . . . . . . 15
|
| 44 | 18, 43 | oveq12d 5550 |
. . . . . . . . . . . . . 14
|
| 45 | 35, 36, 39 | adddid 7143 |
. . . . . . . . . . . . . . 15
|
| 46 | 45 | oveq2d 5548 |
. . . . . . . . . . . . . 14
|
| 47 | 41, 44, 46 | 3eqtr4d 2123 |
. . . . . . . . . . . . 13
|
| 48 | 13 | recnd 7147 |
. . . . . . . . . . . . . . 15
|
| 49 | 15 | recnd 7147 |
. . . . . . . . . . . . . . . 16
|
| 50 | 35, 49 | mulcld 7139 |
. . . . . . . . . . . . . . 15
|
| 51 | 48, 50, 38, 40 | add4d 7277 |
. . . . . . . . . . . . . 14
|
| 52 | 19, 43 | oveq12d 5550 |
. . . . . . . . . . . . . 14
|
| 53 | 35, 49, 39 | adddid 7143 |
. . . . . . . . . . . . . . 15
|
| 54 | 53 | oveq2d 5548 |
. . . . . . . . . . . . . 14
|
| 55 | 51, 52, 54 | 3eqtr4d 2123 |
. . . . . . . . . . . . 13
|
| 56 | 47, 55 | breq12d 3798 |
. . . . . . . . . . . 12
#
# |
| 57 | reapadd1 7696 |
. . . . . . . . . . . . . 14
#
# | |
| 58 | 10, 13, 23, 57 | syl3anc 1169 |
. . . . . . . . . . . . 13
#
# |
| 59 | reapadd1 7696 |
. . . . . . . . . . . . . 14
#
# | |
| 60 | 11, 15, 27, 59 | syl3anc 1169 |
. . . . . . . . . . . . 13
#
# |
| 61 | 58, 60 | orbi12d 739 |
. . . . . . . . . . . 12
# #
# # |
| 62 | 32, 56, 61 | 3bitr4d 218 |
. . . . . . . . . . 11
#
#
# |
| 63 | 17, 20, 62 | 3bitr4d 218 |
. . . . . . . . . 10
#
#
|
| 64 | 63 | ex 113 |
. . . . . . . . 9
#
#
|
| 65 | 64 | rexlimdvva 2484 |
. . . . . . . 8
#
#
|
| 66 | 9, 65 | mpd 13 |
. . . . . . 7
#
#
|
| 67 | 66 | ex 113 |
. . . . . 6
#
#
|
| 68 | 67 | rexlimdvva 2484 |
. . . . 5
#
#
|
| 69 | 5, 68 | mpd 13 |
. . . 4
#
#
|
| 70 | 69 | ex 113 |
. . 3
#
#
|
| 71 | 70 | rexlimdvva 2484 |
. 2
#
#
|
| 72 | 2, 71 | mpd 13 |
1
#
#
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 102
wb 103 wo 661 w3a 919 wceq 1284 wcel 1433 wrex 2349 class class class wbr 3785
(class class class)co 5532 cc 6979 cr 6980 ci 6983 caddc 6984 cmul 6986 # cap 7681 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 |
| This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-ltxr 7158 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 |
| This theorem is referenced by: apadd2 7709 addext 7710 subap0d 7740 |
| Copyright terms: Public domain | W3C validator |