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| Mirrors > Home > ILE Home > Th. List > apadd1 | Unicode version | ||
| Description: Addition respects apartness. Analogue of addcan 8111 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 7928 |
. . 3
     | |
| 2 | 1 | 3ad2ant3 1020 |
. 2
|
| 3 | cnre 7928 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1019 |
. . . . . 6
|
| 5 | 4 | ad2antrr 488 |
. . . . 5
|
| 6 | cnre 7928 |
. . . . . . . . . . 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 8534 |
. . . . . . . . . . . 12
#
# 
# | |
| 17 | 10, 11, 13, 15, 16 | syl22anc 1239 |
. . . . . . . . . . 11
#
#
# |
| 18 | simpr 110 |
. . . . . . . . . . . 12
| |
| 19 | simpllr 534 |
. . . . . . . . . . . 12
| |
| 20 | 18, 19 | breq12d 4011 |
. . . . . . . . . . 11
#
# |
| 21 | simprl 529 |
. . . . . . . . . . . . . . . 16
| |
| 22 | 21 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
|
| 23 | 22 | ad3antrrr 492 |
. . . . . . . . . . . . . 14
|
| 24 | 10, 23 | readdcld 7961 |
. . . . . . . . . . . . 13
|
| 25 | simprr 531 |
. . . . . . . . . . . . . . . 16
| |
| 26 | 25 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
|
| 27 | 26 | ad3antrrr 492 |
. . . . . . . . . . . . . 14
|
| 28 | 11, 27 | readdcld 7961 |
. . . . . . . . . . . . 13
|
| 29 | 13, 23 | readdcld 7961 |
. . . . . . . . . . . . 13
|
| 30 | 15, 27 | readdcld 7961 |
. . . . . . . . . . . . 13
|
| 31 | apreim 8534 |
. . . . . . . . . . . . 13
#
# # | |
| 32 | 24, 28, 29, 30, 31 | syl22anc 1239 |
. . . . . . . . . . . 12
#
# # |
| 33 | 10 | recnd 7960 |
. . . . . . . . . . . . . . 15
|
| 34 | ax-icn 7881 |
. . . . . . . . . . . . . . . . 17
| |
| 35 | 34 | a1i 9 |
. . . . . . . . . . . . . . . 16
|
| 36 | 11 | recnd 7960 |
. . . . . . . . . . . . . . . 16
|
| 37 | 35, 36 | mulcld 7952 |
. . . . . . . . . . . . . . 15
|
| 38 | 23 | recnd 7960 |
. . . . . . . . . . . . . . 15
|
| 39 | 27 | recnd 7960 |
. . . . . . . . . . . . . . . 16
|
| 40 | 35, 39 | mulcld 7952 |
. . . . . . . . . . . . . . 15
|
| 41 | 33, 37, 38, 40 | add4d 8100 |
. . . . . . . . . . . . . 14
|
| 42 | simplr 528 |
. . . . . . . . . . . . . . . 16
| |
| 43 | 42 | ad3antrrr 492 |
. . . . . . . . . . . . . . 15
|
| 44 | 18, 43 | oveq12d 5883 |
. . . . . . . . . . . . . 14
|
| 45 | 35, 36, 39 | adddid 7956 |
. . . . . . . . . . . . . . 15
|
| 46 | 45 | oveq2d 5881 |
. . . . . . . . . . . . . 14
|
| 47 | 41, 44, 46 | 3eqtr4d 2218 |
. . . . . . . . . . . . 13
|
| 48 | 13 | recnd 7960 |
. . . . . . . . . . . . . . 15
|
| 49 | 15 | recnd 7960 |
. . . . . . . . . . . . . . . 16
|
| 50 | 35, 49 | mulcld 7952 |
. . . . . . . . . . . . . . 15
|
| 51 | 48, 50, 38, 40 | add4d 8100 |
. . . . . . . . . . . . . 14
|
| 52 | 19, 43 | oveq12d 5883 |
. . . . . . . . . . . . . 14
|
| 53 | 35, 49, 39 | adddid 7956 |
. . . . . . . . . . . . . . 15
|
| 54 | 53 | oveq2d 5881 |
. . . . . . . . . . . . . 14
|
| 55 | 51, 52, 54 | 3eqtr4d 2218 |
. . . . . . . . . . . . 13
|
| 56 | 47, 55 | breq12d 4011 |
. . . . . . . . . . . 12
#
# |
| 57 | reapadd1 8527 |
. . . . . . . . . . . . . 14
#
# | |
| 58 | 10, 13, 23, 57 | syl3anc 1238 |
. . . . . . . . . . . . 13
#
# |
| 59 | reapadd1 8527 |
. . . . . . . . . . . . . 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 2600 |
. . . . . . . 8
#
#
|
| 66 | 9, 65 | mpd 13 |
. . . . . . 7
#
#
|
| 67 | 66 | ex 115 |
. . . . . 6
#
#
|
| 68 | 67 | rexlimdvva 2600 |
. . . . 5
#
#
|
| 69 | 5, 68 | mpd 13 |
. . . 4
#
#
|
| 70 | 69 | ex 115 |
. . 3
#
#
|
| 71 | 70 | rexlimdvva 2600 |
. 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 2146 wrex 2454 class class class wbr 3998
(class class class)co 5865 cc 7784 cr 7785 ci 7788 caddc 7789 cmul 7791 # cap 8512 |
| 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 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-iota 5170 df-fun 5210 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-pnf 7968 df-mnf 7969 df-ltxr 7971 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 |
| This theorem is referenced by: apadd2 8540 addext 8541 apsub1 8573 subap0 8574 |
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