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| Mirrors > Home > ILE Home > Th. List > apadd1 | Unicode version | ||
| Description: Addition respects apartness. Analogue of addcan 7723 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 7545 |
. . 3
     | |
| 2 | 1 | 3ad2ant3 967 |
. 2
|
| 3 | cnre 7545 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 966 |
. . . . . 6
|
| 5 | 4 | ad2antrr 473 |
. . . . 5
|
| 6 | cnre 7545 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 965 |
. . . . . . . . . 10
|
| 8 | 7 | ad2antrr 473 |
. . . . . . . . 9
|
| 9 | 8 | ad2antrr 473 |
. . . . . . . 8
|
| 10 | simplrl 503 |
. . . . . . . . . . . 12
| |
| 11 | simplrr 504 |
. . . . . . . . . . . 12
| |
| 12 | simprl 499 |
. . . . . . . . . . . . 13
| |
| 13 | 12 | ad3antrrr 477 |
. . . . . . . . . . . 12
|
| 14 | simprr 500 |
. . . . . . . . . . . . 13
| |
| 15 | 14 | ad3antrrr 477 |
. . . . . . . . . . . 12
|
| 16 | apreim 8141 |
. . . . . . . . . . . 12
#
# 
# | |
| 17 | 10, 11, 13, 15, 16 | syl22anc 1176 |
. . . . . . . . . . 11
#
#
# |
| 18 | simpr 109 |
. . . . . . . . . . . 12
| |
| 19 | simpllr 502 |
. . . . . . . . . . . 12
| |
| 20 | 18, 19 | breq12d 3864 |
. . . . . . . . . . 11
#
# |
| 21 | simprl 499 |
. . . . . . . . . . . . . . . 16
| |
| 22 | 21 | ad2antrr 473 |
. . . . . . . . . . . . . . 15
|
| 23 | 22 | ad3antrrr 477 |
. . . . . . . . . . . . . 14
|
| 24 | 10, 23 | readdcld 7578 |
. . . . . . . . . . . . 13
|
| 25 | simprr 500 |
. . . . . . . . . . . . . . . 16
| |
| 26 | 25 | ad2antrr 473 |
. . . . . . . . . . . . . . 15
|
| 27 | 26 | ad3antrrr 477 |
. . . . . . . . . . . . . 14
|
| 28 | 11, 27 | readdcld 7578 |
. . . . . . . . . . . . 13
|
| 29 | 13, 23 | readdcld 7578 |
. . . . . . . . . . . . 13
|
| 30 | 15, 27 | readdcld 7578 |
. . . . . . . . . . . . 13
|
| 31 | apreim 8141 |
. . . . . . . . . . . . 13
#
# # | |
| 32 | 24, 28, 29, 30, 31 | syl22anc 1176 |
. . . . . . . . . . . 12
#
# # |
| 33 | 10 | recnd 7577 |
. . . . . . . . . . . . . . 15
|
| 34 | ax-icn 7501 |
. . . . . . . . . . . . . . . . 17
| |
| 35 | 34 | a1i 9 |
. . . . . . . . . . . . . . . 16
|
| 36 | 11 | recnd 7577 |
. . . . . . . . . . . . . . . 16
|
| 37 | 35, 36 | mulcld 7569 |
. . . . . . . . . . . . . . 15
|
| 38 | 23 | recnd 7577 |
. . . . . . . . . . . . . . 15
|
| 39 | 27 | recnd 7577 |
. . . . . . . . . . . . . . . 16
|
| 40 | 35, 39 | mulcld 7569 |
. . . . . . . . . . . . . . 15
|
| 41 | 33, 37, 38, 40 | add4d 7712 |
. . . . . . . . . . . . . 14
|
| 42 | simplr 498 |
. . . . . . . . . . . . . . . 16
| |
| 43 | 42 | ad3antrrr 477 |
. . . . . . . . . . . . . . 15
|
| 44 | 18, 43 | oveq12d 5684 |
. . . . . . . . . . . . . 14
|
| 45 | 35, 36, 39 | adddid 7573 |
. . . . . . . . . . . . . . 15
|
| 46 | 45 | oveq2d 5682 |
. . . . . . . . . . . . . 14
|
| 47 | 41, 44, 46 | 3eqtr4d 2131 |
. . . . . . . . . . . . 13
|
| 48 | 13 | recnd 7577 |
. . . . . . . . . . . . . . 15
|
| 49 | 15 | recnd 7577 |
. . . . . . . . . . . . . . . 16
|
| 50 | 35, 49 | mulcld 7569 |
. . . . . . . . . . . . . . 15
|
| 51 | 48, 50, 38, 40 | add4d 7712 |
. . . . . . . . . . . . . 14
|
| 52 | 19, 43 | oveq12d 5684 |
. . . . . . . . . . . . . 14
|
| 53 | 35, 49, 39 | adddid 7573 |
. . . . . . . . . . . . . . 15
|
| 54 | 53 | oveq2d 5682 |
. . . . . . . . . . . . . 14
|
| 55 | 51, 52, 54 | 3eqtr4d 2131 |
. . . . . . . . . . . . 13
|
| 56 | 47, 55 | breq12d 3864 |
. . . . . . . . . . . 12
#
# |
| 57 | reapadd1 8134 |
. . . . . . . . . . . . . 14
#
# | |
| 58 | 10, 13, 23, 57 | syl3anc 1175 |
. . . . . . . . . . . . 13
#
# |
| 59 | reapadd1 8134 |
. . . . . . . . . . . . . 14
#
# | |
| 60 | 11, 15, 27, 59 | syl3anc 1175 |
. . . . . . . . . . . . 13
#
# |
| 61 | 58, 60 | orbi12d 743 |
. . . . . . . . . . . 12
# #
# # |
| 62 | 32, 56, 61 | 3bitr4d 219 |
. . . . . . . . . . 11
#
#
# |
| 63 | 17, 20, 62 | 3bitr4d 219 |
. . . . . . . . . 10
#
#
|
| 64 | 63 | ex 114 |
. . . . . . . . 9
#
#
|
| 65 | 64 | rexlimdvva 2497 |
. . . . . . . 8
#
#
|
| 66 | 9, 65 | mpd 13 |
. . . . . . 7
#
#
|
| 67 | 66 | ex 114 |
. . . . . 6
#
#
|
| 68 | 67 | rexlimdvva 2497 |
. . . . 5
#
#
|
| 69 | 5, 68 | mpd 13 |
. . . 4
#
#
|
| 70 | 69 | ex 114 |
. . 3
#
#
|
| 71 | 70 | rexlimdvva 2497 |
. 2
#
#
|
| 72 | 2, 71 | mpd 13 |
1
#
#
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 wo 665 w3a 925 wceq 1290 wcel 1439 wrex 2361 class class class wbr 3851
(class class class)co 5666 cc 7409 cr 7410 ci 7413 caddc 7414 cmul 7416 # cap 8119 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-mulrcl 7505 ax-addcom 7506 ax-mulcom 7507 ax-addass 7508 ax-mulass 7509 ax-distr 7510 ax-i2m1 7511 ax-0lt1 7512 ax-1rid 7513 ax-0id 7514 ax-rnegex 7515 ax-precex 7516 ax-cnre 7517 ax-pre-ltirr 7518 ax-pre-lttrn 7520 ax-pre-apti 7521 ax-pre-ltadd 7522 ax-pre-mulgt0 7523 |
| This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-pnf 7585 df-mnf 7586 df-ltxr 7588 df-sub 7716 df-neg 7717 df-reap 8113 df-ap 8120 |
| This theorem is referenced by: apadd2 8147 addext 8148 apsub1 8178 subap0 8179 subap0d 8180 |
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