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| Mirrors > Home > ILE Home > Th. List > apadd1 | Unicode version | ||
| Description: Addition respects apartness. Analogue of addcan 8167 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 7983 |
. . 3
     | |
| 2 | 1 | 3ad2ant3 1022 |
. 2
|
| 3 | cnre 7983 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1021 |
. . . . . 6
|
| 5 | 4 | ad2antrr 488 |
. . . . 5
|
| 6 | cnre 7983 |
. . . . . . . . . . 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 8590 |
. . . . . . . . . . . 12
#
# 
# | |
| 17 | 10, 11, 13, 15, 16 | syl22anc 1250 |
. . . . . . . . . . 11
#
#
# |
| 18 | simpr 110 |
. . . . . . . . . . . 12
| |
| 19 | simpllr 534 |
. . . . . . . . . . . 12
| |
| 20 | 18, 19 | breq12d 4031 |
. . . . . . . . . . 11
#
# |
| 21 | simprl 529 |
. . . . . . . . . . . . . . . 16
| |
| 22 | 21 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
|
| 23 | 22 | ad3antrrr 492 |
. . . . . . . . . . . . . 14
|
| 24 | 10, 23 | readdcld 8017 |
. . . . . . . . . . . . 13
|
| 25 | simprr 531 |
. . . . . . . . . . . . . . . 16
| |
| 26 | 25 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
|
| 27 | 26 | ad3antrrr 492 |
. . . . . . . . . . . . . 14
|
| 28 | 11, 27 | readdcld 8017 |
. . . . . . . . . . . . 13
|
| 29 | 13, 23 | readdcld 8017 |
. . . . . . . . . . . . 13
|
| 30 | 15, 27 | readdcld 8017 |
. . . . . . . . . . . . 13
|
| 31 | apreim 8590 |
. . . . . . . . . . . . 13
#
# # | |
| 32 | 24, 28, 29, 30, 31 | syl22anc 1250 |
. . . . . . . . . . . 12
#
# # |
| 33 | 10 | recnd 8016 |
. . . . . . . . . . . . . . 15
|
| 34 | ax-icn 7936 |
. . . . . . . . . . . . . . . . 17
| |
| 35 | 34 | a1i 9 |
. . . . . . . . . . . . . . . 16
|
| 36 | 11 | recnd 8016 |
. . . . . . . . . . . . . . . 16
|
| 37 | 35, 36 | mulcld 8008 |
. . . . . . . . . . . . . . 15
|
| 38 | 23 | recnd 8016 |
. . . . . . . . . . . . . . 15
|
| 39 | 27 | recnd 8016 |
. . . . . . . . . . . . . . . 16
|
| 40 | 35, 39 | mulcld 8008 |
. . . . . . . . . . . . . . 15
|
| 41 | 33, 37, 38, 40 | add4d 8156 |
. . . . . . . . . . . . . 14
|
| 42 | simplr 528 |
. . . . . . . . . . . . . . . 16
| |
| 43 | 42 | ad3antrrr 492 |
. . . . . . . . . . . . . . 15
|
| 44 | 18, 43 | oveq12d 5914 |
. . . . . . . . . . . . . 14
|
| 45 | 35, 36, 39 | adddid 8012 |
. . . . . . . . . . . . . . 15
|
| 46 | 45 | oveq2d 5912 |
. . . . . . . . . . . . . 14
|
| 47 | 41, 44, 46 | 3eqtr4d 2232 |
. . . . . . . . . . . . 13
|
| 48 | 13 | recnd 8016 |
. . . . . . . . . . . . . . 15
|
| 49 | 15 | recnd 8016 |
. . . . . . . . . . . . . . . 16
|
| 50 | 35, 49 | mulcld 8008 |
. . . . . . . . . . . . . . 15
|
| 51 | 48, 50, 38, 40 | add4d 8156 |
. . . . . . . . . . . . . 14
|
| 52 | 19, 43 | oveq12d 5914 |
. . . . . . . . . . . . . 14
|
| 53 | 35, 49, 39 | adddid 8012 |
. . . . . . . . . . . . . . 15
|
| 54 | 53 | oveq2d 5912 |
. . . . . . . . . . . . . 14
|
| 55 | 51, 52, 54 | 3eqtr4d 2232 |
. . . . . . . . . . . . 13
|
| 56 | 47, 55 | breq12d 4031 |
. . . . . . . . . . . 12
#
# |
| 57 | reapadd1 8583 |
. . . . . . . . . . . . . 14
#
# | |
| 58 | 10, 13, 23, 57 | syl3anc 1249 |
. . . . . . . . . . . . 13
#
# |
| 59 | reapadd1 8583 |
. . . . . . . . . . . . . 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 2615 |
. . . . . . . 8
#
#
|
| 66 | 9, 65 | mpd 13 |
. . . . . . 7
#
#
|
| 67 | 66 | ex 115 |
. . . . . 6
#
#
|
| 68 | 67 | rexlimdvva 2615 |
. . . . 5
#
#
|
| 69 | 5, 68 | mpd 13 |
. . . 4
#
#
|
| 70 | 69 | ex 115 |
. . 3
#
#
|
| 71 | 70 | rexlimdvva 2615 |
. 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 2160 wrex 2469 class class class wbr 4018
(class class class)co 5896 cc 7839 cr 7840 ci 7843 caddc 7844 cmul 7846 # cap 8568 |
| 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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-mulrcl 7940 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-1rid 7948 ax-0id 7949 ax-rnegex 7950 ax-precex 7951 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-lttrn 7955 ax-pre-apti 7956 ax-pre-ltadd 7957 ax-pre-mulgt0 7958 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-pnf 8024 df-mnf 8025 df-ltxr 8027 df-sub 8160 df-neg 8161 df-reap 8562 df-ap 8569 |
| This theorem is referenced by: apadd2 8596 addext 8597 apsub1 8629 subap0 8630 |
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