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| Mirrors > Home > ILE Home > Th. List > apadd1 | Unicode version | ||
| Description: Addition respects apartness. Analogue of addcan 8099 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 7916 |
. . 3
     | |
| 2 | 1 | 3ad2ant3 1015 |
. 2
|
| 3 | cnre 7916 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1014 |
. . . . . 6
|
| 5 | 4 | ad2antrr 485 |
. . . . 5
|
| 6 | cnre 7916 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 1013 |
. . . . . . . . . 10
|
| 8 | 7 | ad2antrr 485 |
. . . . . . . . 9
|
| 9 | 8 | ad2antrr 485 |
. . . . . . . 8
|
| 10 | simplrl 530 |
. . . . . . . . . . . 12
| |
| 11 | simplrr 531 |
. . . . . . . . . . . 12
| |
| 12 | simprl 526 |
. . . . . . . . . . . . 13
| |
| 13 | 12 | ad3antrrr 489 |
. . . . . . . . . . . 12
|
| 14 | simprr 527 |
. . . . . . . . . . . . 13
| |
| 15 | 14 | ad3antrrr 489 |
. . . . . . . . . . . 12
|
| 16 | apreim 8522 |
. . . . . . . . . . . 12
#
# 
# | |
| 17 | 10, 11, 13, 15, 16 | syl22anc 1234 |
. . . . . . . . . . 11
#
#
# |
| 18 | simpr 109 |
. . . . . . . . . . . 12
| |
| 19 | simpllr 529 |
. . . . . . . . . . . 12
| |
| 20 | 18, 19 | breq12d 4002 |
. . . . . . . . . . 11
#
# |
| 21 | simprl 526 |
. . . . . . . . . . . . . . . 16
| |
| 22 | 21 | ad2antrr 485 |
. . . . . . . . . . . . . . 15
|
| 23 | 22 | ad3antrrr 489 |
. . . . . . . . . . . . . 14
|
| 24 | 10, 23 | readdcld 7949 |
. . . . . . . . . . . . 13
|
| 25 | simprr 527 |
. . . . . . . . . . . . . . . 16
| |
| 26 | 25 | ad2antrr 485 |
. . . . . . . . . . . . . . 15
|
| 27 | 26 | ad3antrrr 489 |
. . . . . . . . . . . . . 14
|
| 28 | 11, 27 | readdcld 7949 |
. . . . . . . . . . . . 13
|
| 29 | 13, 23 | readdcld 7949 |
. . . . . . . . . . . . 13
|
| 30 | 15, 27 | readdcld 7949 |
. . . . . . . . . . . . 13
|
| 31 | apreim 8522 |
. . . . . . . . . . . . 13
#
# # | |
| 32 | 24, 28, 29, 30, 31 | syl22anc 1234 |
. . . . . . . . . . . 12
#
# # |
| 33 | 10 | recnd 7948 |
. . . . . . . . . . . . . . 15
|
| 34 | ax-icn 7869 |
. . . . . . . . . . . . . . . . 17
| |
| 35 | 34 | a1i 9 |
. . . . . . . . . . . . . . . 16
|
| 36 | 11 | recnd 7948 |
. . . . . . . . . . . . . . . 16
|
| 37 | 35, 36 | mulcld 7940 |
. . . . . . . . . . . . . . 15
|
| 38 | 23 | recnd 7948 |
. . . . . . . . . . . . . . 15
|
| 39 | 27 | recnd 7948 |
. . . . . . . . . . . . . . . 16
|
| 40 | 35, 39 | mulcld 7940 |
. . . . . . . . . . . . . . 15
|
| 41 | 33, 37, 38, 40 | add4d 8088 |
. . . . . . . . . . . . . 14
|
| 42 | simplr 525 |
. . . . . . . . . . . . . . . 16
| |
| 43 | 42 | ad3antrrr 489 |
. . . . . . . . . . . . . . 15
|
| 44 | 18, 43 | oveq12d 5871 |
. . . . . . . . . . . . . 14
|
| 45 | 35, 36, 39 | adddid 7944 |
. . . . . . . . . . . . . . 15
|
| 46 | 45 | oveq2d 5869 |
. . . . . . . . . . . . . 14
|
| 47 | 41, 44, 46 | 3eqtr4d 2213 |
. . . . . . . . . . . . 13
|
| 48 | 13 | recnd 7948 |
. . . . . . . . . . . . . . 15
|
| 49 | 15 | recnd 7948 |
. . . . . . . . . . . . . . . 16
|
| 50 | 35, 49 | mulcld 7940 |
. . . . . . . . . . . . . . 15
|
| 51 | 48, 50, 38, 40 | add4d 8088 |
. . . . . . . . . . . . . 14
|
| 52 | 19, 43 | oveq12d 5871 |
. . . . . . . . . . . . . 14
|
| 53 | 35, 49, 39 | adddid 7944 |
. . . . . . . . . . . . . . 15
|
| 54 | 53 | oveq2d 5869 |
. . . . . . . . . . . . . 14
|
| 55 | 51, 52, 54 | 3eqtr4d 2213 |
. . . . . . . . . . . . 13
|
| 56 | 47, 55 | breq12d 4002 |
. . . . . . . . . . . 12
#
# |
| 57 | reapadd1 8515 |
. . . . . . . . . . . . . 14
#
# | |
| 58 | 10, 13, 23, 57 | syl3anc 1233 |
. . . . . . . . . . . . 13
#
# |
| 59 | reapadd1 8515 |
. . . . . . . . . . . . . 14
#
# | |
| 60 | 11, 15, 27, 59 | syl3anc 1233 |
. . . . . . . . . . . . 13
#
# |
| 61 | 58, 60 | orbi12d 788 |
. . . . . . . . . . . 12
# #
# # |
| 62 | 32, 56, 61 | 3bitr4d 219 |
. . . . . . . . . . 11
#
#
# |
| 63 | 17, 20, 62 | 3bitr4d 219 |
. . . . . . . . . 10
#
#
|
| 64 | 63 | ex 114 |
. . . . . . . . 9
#
#
|
| 65 | 64 | rexlimdvva 2595 |
. . . . . . . 8
#
#
|
| 66 | 9, 65 | mpd 13 |
. . . . . . 7
#
#
|
| 67 | 66 | ex 114 |
. . . . . 6
#
#
|
| 68 | 67 | rexlimdvva 2595 |
. . . . 5
#
#
|
| 69 | 5, 68 | mpd 13 |
. . . 4
#
#
|
| 70 | 69 | ex 114 |
. . 3
#
#
|
| 71 | 70 | rexlimdvva 2595 |
. 2
#
#
|
| 72 | 2, 71 | mpd 13 |
1
#
#
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 wo 703 w3a 973 wceq 1348 wcel 2141 wrex 2449 class class class wbr 3989
(class class class)co 5853 cc 7772 cr 7773 ci 7776 caddc 7777 cmul 7779 # cap 8500 |
| 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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 |
| This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-ltxr 7959 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 |
| This theorem is referenced by: apadd2 8528 addext 8529 apsub1 8561 subap0 8562 |
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