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| Mirrors > Home > ILE Home > Th. List > apadd1 | Unicode version | ||
| Description: Addition respects apartness. Analogue of addcan 8206 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 8022 |
. . 3
     | |
| 2 | 1 | 3ad2ant3 1022 |
. 2
|
| 3 | cnre 8022 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1021 |
. . . . . 6
|
| 5 | 4 | ad2antrr 488 |
. . . . 5
|
| 6 | cnre 8022 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 1020 |
. . . . . . . . . 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 8630 |
. . . . . . . . . . . 12
#
# 
# | |
| 17 | 10, 11, 13, 15, 16 | syl22anc 1250 |
. . . . . . . . . . 11
#
#
# |
| 18 | simpr 110 |
. . . . . . . . . . . 12
| |
| 19 | simpllr 534 |
. . . . . . . . . . . 12
| |
| 20 | 18, 19 | breq12d 4046 |
. . . . . . . . . . 11
#
# |
| 21 | simprl 529 |
. . . . . . . . . . . . . . . 16
| |
| 22 | 21 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
|
| 23 | 22 | ad3antrrr 492 |
. . . . . . . . . . . . . 14
|
| 24 | 10, 23 | readdcld 8056 |
. . . . . . . . . . . . 13
|
| 25 | simprr 531 |
. . . . . . . . . . . . . . . 16
| |
| 26 | 25 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
|
| 27 | 26 | ad3antrrr 492 |
. . . . . . . . . . . . . 14
|
| 28 | 11, 27 | readdcld 8056 |
. . . . . . . . . . . . 13
|
| 29 | 13, 23 | readdcld 8056 |
. . . . . . . . . . . . 13
|
| 30 | 15, 27 | readdcld 8056 |
. . . . . . . . . . . . 13
|
| 31 | apreim 8630 |
. . . . . . . . . . . . 13
#
# # | |
| 32 | 24, 28, 29, 30, 31 | syl22anc 1250 |
. . . . . . . . . . . 12
#
# # |
| 33 | 10 | recnd 8055 |
. . . . . . . . . . . . . . 15
|
| 34 | ax-icn 7974 |
. . . . . . . . . . . . . . . . 17
| |
| 35 | 34 | a1i 9 |
. . . . . . . . . . . . . . . 16
|
| 36 | 11 | recnd 8055 |
. . . . . . . . . . . . . . . 16
|
| 37 | 35, 36 | mulcld 8047 |
. . . . . . . . . . . . . . 15
|
| 38 | 23 | recnd 8055 |
. . . . . . . . . . . . . . 15
|
| 39 | 27 | recnd 8055 |
. . . . . . . . . . . . . . . 16
|
| 40 | 35, 39 | mulcld 8047 |
. . . . . . . . . . . . . . 15
|
| 41 | 33, 37, 38, 40 | add4d 8195 |
. . . . . . . . . . . . . 14
|
| 42 | simplr 528 |
. . . . . . . . . . . . . . . 16
| |
| 43 | 42 | ad3antrrr 492 |
. . . . . . . . . . . . . . 15
|
| 44 | 18, 43 | oveq12d 5940 |
. . . . . . . . . . . . . 14
|
| 45 | 35, 36, 39 | adddid 8051 |
. . . . . . . . . . . . . . 15
|
| 46 | 45 | oveq2d 5938 |
. . . . . . . . . . . . . 14
|
| 47 | 41, 44, 46 | 3eqtr4d 2239 |
. . . . . . . . . . . . 13
|
| 48 | 13 | recnd 8055 |
. . . . . . . . . . . . . . 15
|
| 49 | 15 | recnd 8055 |
. . . . . . . . . . . . . . . 16
|
| 50 | 35, 49 | mulcld 8047 |
. . . . . . . . . . . . . . 15
|
| 51 | 48, 50, 38, 40 | add4d 8195 |
. . . . . . . . . . . . . 14
|
| 52 | 19, 43 | oveq12d 5940 |
. . . . . . . . . . . . . 14
|
| 53 | 35, 49, 39 | adddid 8051 |
. . . . . . . . . . . . . . 15
|
| 54 | 53 | oveq2d 5938 |
. . . . . . . . . . . . . 14
|
| 55 | 51, 52, 54 | 3eqtr4d 2239 |
. . . . . . . . . . . . 13
|
| 56 | 47, 55 | breq12d 4046 |
. . . . . . . . . . . 12
#
# |
| 57 | reapadd1 8623 |
. . . . . . . . . . . . . 14
#
# | |
| 58 | 10, 13, 23, 57 | syl3anc 1249 |
. . . . . . . . . . . . 13
#
# |
| 59 | reapadd1 8623 |
. . . . . . . . . . . . . 14
#
# | |
| 60 | 11, 15, 27, 59 | syl3anc 1249 |
. . . . . . . . . . . . 13
#
# |
| 61 | 58, 60 | orbi12d 794 |
. . . . . . . . . . . 12
# #
# # |
| 62 | 32, 56, 61 | 3bitr4d 220 |
. . . . . . . . . . 11
#
#
# |
| 63 | 17, 20, 62 | 3bitr4d 220 |
. . . . . . . . . 10
#
#
|
| 64 | 63 | ex 115 |
. . . . . . . . 9
#
#
|
| 65 | 64 | rexlimdvva 2622 |
. . . . . . . 8
#
#
|
| 66 | 9, 65 | mpd 13 |
. . . . . . 7
#
#
|
| 67 | 66 | ex 115 |
. . . . . 6
#
#
|
| 68 | 67 | rexlimdvva 2622 |
. . . . 5
#
#
|
| 69 | 5, 68 | mpd 13 |
. . . 4
#
#
|
| 70 | 69 | ex 115 |
. . 3
#
#
|
| 71 | 70 | rexlimdvva 2622 |
. 2
#
#
|
| 72 | 2, 71 | mpd 13 |
1
#
#
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wo 709 w3a 980 wceq 1364 wcel 2167 wrex 2476 class class class wbr 4033
(class class class)co 5922 cc 7877 cr 7878 ci 7881 caddc 7882 cmul 7884 # cap 8608 |
| 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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-br 4034 df-opab 4095 df-id 4328 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-iota 5219 df-fun 5260 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-pnf 8063 df-mnf 8064 df-ltxr 8066 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 |
| This theorem is referenced by: apadd2 8636 addext 8637 apsub1 8669 subap0 8670 |
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