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| Mirrors > Home > ILE Home > Th. List > apadd1 | Unicode version | ||
| Description: Addition respects apartness. Analogue of addcan 8418 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 8235 |
. . 3
     | |
| 2 | 1 | 3ad2ant3 1047 |
. 2
|
| 3 | cnre 8235 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1046 |
. . . . . 6
|
| 5 | 4 | ad2antrr 488 |
. . . . 5
|
| 6 | cnre 8235 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 1045 |
. . . . . . . . . 10
|
| 8 | 7 | ad2antrr 488 |
. . . . . . . . 9
|
| 9 | 8 | ad2antrr 488 |
. . . . . . . 8
|
| 10 | simplrl 537 |
. . . . . . . . . . . 12
| |
| 11 | simplrr 538 |
. . . . . . . . . . . 12
| |
| 12 | simprl 531 |
. . . . . . . . . . . . 13
| |
| 13 | 12 | ad3antrrr 492 |
. . . . . . . . . . . 12
|
| 14 | simprr 533 |
. . . . . . . . . . . . 13
| |
| 15 | 14 | ad3antrrr 492 |
. . . . . . . . . . . 12
|
| 16 | apreim 8842 |
. . . . . . . . . . . 12
#
# 
# | |
| 17 | 10, 11, 13, 15, 16 | syl22anc 1275 |
. . . . . . . . . . 11
#
#
# |
| 18 | simpr 110 |
. . . . . . . . . . . 12
| |
| 19 | simpllr 536 |
. . . . . . . . . . . 12
| |
| 20 | 18, 19 | breq12d 4106 |
. . . . . . . . . . 11
#
# |
| 21 | simprl 531 |
. . . . . . . . . . . . . . . 16
| |
| 22 | 21 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
|
| 23 | 22 | ad3antrrr 492 |
. . . . . . . . . . . . . 14
|
| 24 | 10, 23 | readdcld 8268 |
. . . . . . . . . . . . 13
|
| 25 | simprr 533 |
. . . . . . . . . . . . . . . 16
| |
| 26 | 25 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
|
| 27 | 26 | ad3antrrr 492 |
. . . . . . . . . . . . . 14
|
| 28 | 11, 27 | readdcld 8268 |
. . . . . . . . . . . . 13
|
| 29 | 13, 23 | readdcld 8268 |
. . . . . . . . . . . . 13
|
| 30 | 15, 27 | readdcld 8268 |
. . . . . . . . . . . . 13
|
| 31 | apreim 8842 |
. . . . . . . . . . . . 13
#
# # | |
| 32 | 24, 28, 29, 30, 31 | syl22anc 1275 |
. . . . . . . . . . . 12
#
# # |
| 33 | 10 | recnd 8267 |
. . . . . . . . . . . . . . 15
|
| 34 | ax-icn 8187 |
. . . . . . . . . . . . . . . . 17
| |
| 35 | 34 | a1i 9 |
. . . . . . . . . . . . . . . 16
|
| 36 | 11 | recnd 8267 |
. . . . . . . . . . . . . . . 16
|
| 37 | 35, 36 | mulcld 8259 |
. . . . . . . . . . . . . . 15
|
| 38 | 23 | recnd 8267 |
. . . . . . . . . . . . . . 15
|
| 39 | 27 | recnd 8267 |
. . . . . . . . . . . . . . . 16
|
| 40 | 35, 39 | mulcld 8259 |
. . . . . . . . . . . . . . 15
|
| 41 | 33, 37, 38, 40 | add4d 8407 |
. . . . . . . . . . . . . 14
|
| 42 | simplr 529 |
. . . . . . . . . . . . . . . 16
| |
| 43 | 42 | ad3antrrr 492 |
. . . . . . . . . . . . . . 15
|
| 44 | 18, 43 | oveq12d 6046 |
. . . . . . . . . . . . . 14
|
| 45 | 35, 36, 39 | adddid 8263 |
. . . . . . . . . . . . . . 15
|
| 46 | 45 | oveq2d 6044 |
. . . . . . . . . . . . . 14
|
| 47 | 41, 44, 46 | 3eqtr4d 2274 |
. . . . . . . . . . . . 13
|
| 48 | 13 | recnd 8267 |
. . . . . . . . . . . . . . 15
|
| 49 | 15 | recnd 8267 |
. . . . . . . . . . . . . . . 16
|
| 50 | 35, 49 | mulcld 8259 |
. . . . . . . . . . . . . . 15
|
| 51 | 48, 50, 38, 40 | add4d 8407 |
. . . . . . . . . . . . . 14
|
| 52 | 19, 43 | oveq12d 6046 |
. . . . . . . . . . . . . 14
|
| 53 | 35, 49, 39 | adddid 8263 |
. . . . . . . . . . . . . . 15
|
| 54 | 53 | oveq2d 6044 |
. . . . . . . . . . . . . 14
|
| 55 | 51, 52, 54 | 3eqtr4d 2274 |
. . . . . . . . . . . . 13
|
| 56 | 47, 55 | breq12d 4106 |
. . . . . . . . . . . 12
#
# |
| 57 | reapadd1 8835 |
. . . . . . . . . . . . . 14
#
# | |
| 58 | 10, 13, 23, 57 | syl3anc 1274 |
. . . . . . . . . . . . 13
#
# |
| 59 | reapadd1 8835 |
. . . . . . . . . . . . . 14
#
# | |
| 60 | 11, 15, 27, 59 | syl3anc 1274 |
. . . . . . . . . . . . 13
#
# |
| 61 | 58, 60 | orbi12d 801 |
. . . . . . . . . . . 12
# #
# # |
| 62 | 32, 56, 61 | 3bitr4d 220 |
. . . . . . . . . . 11
#
#
# |
| 63 | 17, 20, 62 | 3bitr4d 220 |
. . . . . . . . . 10
#
#
|
| 64 | 63 | ex 115 |
. . . . . . . . 9
#
#
|
| 65 | 64 | rexlimdvva 2659 |
. . . . . . . 8
#
#
|
| 66 | 9, 65 | mpd 13 |
. . . . . . 7
#
#
|
| 67 | 66 | ex 115 |
. . . . . 6
#
#
|
| 68 | 67 | rexlimdvva 2659 |
. . . . 5
#
#
|
| 69 | 5, 68 | mpd 13 |
. . . 4
#
#
|
| 70 | 69 | ex 115 |
. . 3
#
#
|
| 71 | 70 | rexlimdvva 2659 |
. 2
#
#
|
| 72 | 2, 71 | mpd 13 |
1
#
#
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wo 716 w3a 1005 wceq 1398 wcel 2202 wrex 2512 class class class wbr 4093
(class class class)co 6028 cc 8090 cr 8091 ci 8094 caddc 8095 cmul 8097 # cap 8820 |
| 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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-cnex 8183 ax-resscn 8184 ax-1cn 8185 ax-1re 8186 ax-icn 8187 ax-addcl 8188 ax-addrcl 8189 ax-mulcl 8190 ax-mulrcl 8191 ax-addcom 8192 ax-mulcom 8193 ax-addass 8194 ax-mulass 8195 ax-distr 8196 ax-i2m1 8197 ax-0lt1 8198 ax-1rid 8199 ax-0id 8200 ax-rnegex 8201 ax-precex 8202 ax-cnre 8203 ax-pre-ltirr 8204 ax-pre-lttrn 8206 ax-pre-apti 8207 ax-pre-ltadd 8208 ax-pre-mulgt0 8209 |
| This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-br 4094 df-opab 4156 df-id 4396 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-iota 5293 df-fun 5335 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-pnf 8275 df-mnf 8276 df-ltxr 8278 df-sub 8411 df-neg 8412 df-reap 8814 df-ap 8821 |
| This theorem is referenced by: apadd2 8848 addext 8849 apsub1 8881 subap0 8882 |
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