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| Mirrors > Home > ILE Home > Th. List > apadd1 | Unicode version | ||
| Description: Addition respects apartness. Analogue of addcan 7966 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 7786 |
. . 3
     | |
| 2 | 1 | 3ad2ant3 1005 |
. 2
|
| 3 | cnre 7786 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1004 |
. . . . . 6
|
| 5 | 4 | ad2antrr 480 |
. . . . 5
|
| 6 | cnre 7786 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 1003 |
. . . . . . . . . 10
|
| 8 | 7 | ad2antrr 480 |
. . . . . . . . 9
|
| 9 | 8 | ad2antrr 480 |
. . . . . . . 8
|
| 10 | simplrl 525 |
. . . . . . . . . . . 12
| |
| 11 | simplrr 526 |
. . . . . . . . . . . 12
| |
| 12 | simprl 521 |
. . . . . . . . . . . . 13
| |
| 13 | 12 | ad3antrrr 484 |
. . . . . . . . . . . 12
|
| 14 | simprr 522 |
. . . . . . . . . . . . 13
| |
| 15 | 14 | ad3antrrr 484 |
. . . . . . . . . . . 12
|
| 16 | apreim 8389 |
. . . . . . . . . . . 12
#
# 
# | |
| 17 | 10, 11, 13, 15, 16 | syl22anc 1218 |
. . . . . . . . . . 11
#
#
# |
| 18 | simpr 109 |
. . . . . . . . . . . 12
| |
| 19 | simpllr 524 |
. . . . . . . . . . . 12
| |
| 20 | 18, 19 | breq12d 3950 |
. . . . . . . . . . 11
#
# |
| 21 | simprl 521 |
. . . . . . . . . . . . . . . 16
| |
| 22 | 21 | ad2antrr 480 |
. . . . . . . . . . . . . . 15
|
| 23 | 22 | ad3antrrr 484 |
. . . . . . . . . . . . . 14
|
| 24 | 10, 23 | readdcld 7819 |
. . . . . . . . . . . . 13
|
| 25 | simprr 522 |
. . . . . . . . . . . . . . . 16
| |
| 26 | 25 | ad2antrr 480 |
. . . . . . . . . . . . . . 15
|
| 27 | 26 | ad3antrrr 484 |
. . . . . . . . . . . . . 14
|
| 28 | 11, 27 | readdcld 7819 |
. . . . . . . . . . . . 13
|
| 29 | 13, 23 | readdcld 7819 |
. . . . . . . . . . . . 13
|
| 30 | 15, 27 | readdcld 7819 |
. . . . . . . . . . . . 13
|
| 31 | apreim 8389 |
. . . . . . . . . . . . 13
#
# # | |
| 32 | 24, 28, 29, 30, 31 | syl22anc 1218 |
. . . . . . . . . . . 12
#
# # |
| 33 | 10 | recnd 7818 |
. . . . . . . . . . . . . . 15
|
| 34 | ax-icn 7739 |
. . . . . . . . . . . . . . . . 17
| |
| 35 | 34 | a1i 9 |
. . . . . . . . . . . . . . . 16
|
| 36 | 11 | recnd 7818 |
. . . . . . . . . . . . . . . 16
|
| 37 | 35, 36 | mulcld 7810 |
. . . . . . . . . . . . . . 15
|
| 38 | 23 | recnd 7818 |
. . . . . . . . . . . . . . 15
|
| 39 | 27 | recnd 7818 |
. . . . . . . . . . . . . . . 16
|
| 40 | 35, 39 | mulcld 7810 |
. . . . . . . . . . . . . . 15
|
| 41 | 33, 37, 38, 40 | add4d 7955 |
. . . . . . . . . . . . . 14
|
| 42 | simplr 520 |
. . . . . . . . . . . . . . . 16
| |
| 43 | 42 | ad3antrrr 484 |
. . . . . . . . . . . . . . 15
|
| 44 | 18, 43 | oveq12d 5800 |
. . . . . . . . . . . . . 14
|
| 45 | 35, 36, 39 | adddid 7814 |
. . . . . . . . . . . . . . 15
|
| 46 | 45 | oveq2d 5798 |
. . . . . . . . . . . . . 14
|
| 47 | 41, 44, 46 | 3eqtr4d 2183 |
. . . . . . . . . . . . 13
|
| 48 | 13 | recnd 7818 |
. . . . . . . . . . . . . . 15
|
| 49 | 15 | recnd 7818 |
. . . . . . . . . . . . . . . 16
|
| 50 | 35, 49 | mulcld 7810 |
. . . . . . . . . . . . . . 15
|
| 51 | 48, 50, 38, 40 | add4d 7955 |
. . . . . . . . . . . . . 14
|
| 52 | 19, 43 | oveq12d 5800 |
. . . . . . . . . . . . . 14
|
| 53 | 35, 49, 39 | adddid 7814 |
. . . . . . . . . . . . . . 15
|
| 54 | 53 | oveq2d 5798 |
. . . . . . . . . . . . . 14
|
| 55 | 51, 52, 54 | 3eqtr4d 2183 |
. . . . . . . . . . . . 13
|
| 56 | 47, 55 | breq12d 3950 |
. . . . . . . . . . . 12
#
# |
| 57 | reapadd1 8382 |
. . . . . . . . . . . . . 14
#
# | |
| 58 | 10, 13, 23, 57 | syl3anc 1217 |
. . . . . . . . . . . . 13
#
# |
| 59 | reapadd1 8382 |
. . . . . . . . . . . . . 14
#
# | |
| 60 | 11, 15, 27, 59 | syl3anc 1217 |
. . . . . . . . . . . . 13
#
# |
| 61 | 58, 60 | orbi12d 783 |
. . . . . . . . . . . 12
# #
# # |
| 62 | 32, 56, 61 | 3bitr4d 219 |
. . . . . . . . . . 11
#
#
# |
| 63 | 17, 20, 62 | 3bitr4d 219 |
. . . . . . . . . 10
#
#
|
| 64 | 63 | ex 114 |
. . . . . . . . 9
#
#
|
| 65 | 64 | rexlimdvva 2560 |
. . . . . . . 8
#
#
|
| 66 | 9, 65 | mpd 13 |
. . . . . . 7
#
#
|
| 67 | 66 | ex 114 |
. . . . . 6
#
#
|
| 68 | 67 | rexlimdvva 2560 |
. . . . 5
#
#
|
| 69 | 5, 68 | mpd 13 |
. . . 4
#
#
|
| 70 | 69 | ex 114 |
. . 3
#
#
|
| 71 | 70 | rexlimdvva 2560 |
. 2
#
#
|
| 72 | 2, 71 | mpd 13 |
1
#
#
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 wo 698 w3a 963 wceq 1332 wcel 1481 wrex 2418 class class class wbr 3937
(class class class)co 5782 cc 7642 cr 7643 ci 7646 caddc 7647 cmul 7649 # cap 8367 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 |
| This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-ltxr 7829 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 |
| This theorem is referenced by: apadd2 8395 addext 8396 apsub1 8428 subap0 8429 subap0d 8430 |
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