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| Mirrors > Home > ILE Home > Th. List > apadd1 | Unicode version | ||
| Description: Addition respects apartness. Analogue of addcan 8223 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 8039 |
. . 3
     | |
| 2 | 1 | 3ad2ant3 1022 |
. 2
|
| 3 | cnre 8039 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1021 |
. . . . . 6
|
| 5 | 4 | ad2antrr 488 |
. . . . 5
|
| 6 | cnre 8039 |
. . . . . . . . . . 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 8647 |
. . . . . . . . . . . 12
#
# 
# | |
| 17 | 10, 11, 13, 15, 16 | syl22anc 1250 |
. . . . . . . . . . 11
#
#
# |
| 18 | simpr 110 |
. . . . . . . . . . . 12
| |
| 19 | simpllr 534 |
. . . . . . . . . . . 12
| |
| 20 | 18, 19 | breq12d 4047 |
. . . . . . . . . . 11
#
# |
| 21 | simprl 529 |
. . . . . . . . . . . . . . . 16
| |
| 22 | 21 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
|
| 23 | 22 | ad3antrrr 492 |
. . . . . . . . . . . . . 14
|
| 24 | 10, 23 | readdcld 8073 |
. . . . . . . . . . . . 13
|
| 25 | simprr 531 |
. . . . . . . . . . . . . . . 16
| |
| 26 | 25 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
|
| 27 | 26 | ad3antrrr 492 |
. . . . . . . . . . . . . 14
|
| 28 | 11, 27 | readdcld 8073 |
. . . . . . . . . . . . 13
|
| 29 | 13, 23 | readdcld 8073 |
. . . . . . . . . . . . 13
|
| 30 | 15, 27 | readdcld 8073 |
. . . . . . . . . . . . 13
|
| 31 | apreim 8647 |
. . . . . . . . . . . . 13
#
# # | |
| 32 | 24, 28, 29, 30, 31 | syl22anc 1250 |
. . . . . . . . . . . 12
#
# # |
| 33 | 10 | recnd 8072 |
. . . . . . . . . . . . . . 15
|
| 34 | ax-icn 7991 |
. . . . . . . . . . . . . . . . 17
| |
| 35 | 34 | a1i 9 |
. . . . . . . . . . . . . . . 16
|
| 36 | 11 | recnd 8072 |
. . . . . . . . . . . . . . . 16
|
| 37 | 35, 36 | mulcld 8064 |
. . . . . . . . . . . . . . 15
|
| 38 | 23 | recnd 8072 |
. . . . . . . . . . . . . . 15
|
| 39 | 27 | recnd 8072 |
. . . . . . . . . . . . . . . 16
|
| 40 | 35, 39 | mulcld 8064 |
. . . . . . . . . . . . . . 15
|
| 41 | 33, 37, 38, 40 | add4d 8212 |
. . . . . . . . . . . . . 14
|
| 42 | simplr 528 |
. . . . . . . . . . . . . . . 16
| |
| 43 | 42 | ad3antrrr 492 |
. . . . . . . . . . . . . . 15
|
| 44 | 18, 43 | oveq12d 5943 |
. . . . . . . . . . . . . 14
|
| 45 | 35, 36, 39 | adddid 8068 |
. . . . . . . . . . . . . . 15
|
| 46 | 45 | oveq2d 5941 |
. . . . . . . . . . . . . 14
|
| 47 | 41, 44, 46 | 3eqtr4d 2239 |
. . . . . . . . . . . . 13
|
| 48 | 13 | recnd 8072 |
. . . . . . . . . . . . . . 15
|
| 49 | 15 | recnd 8072 |
. . . . . . . . . . . . . . . 16
|
| 50 | 35, 49 | mulcld 8064 |
. . . . . . . . . . . . . . 15
|
| 51 | 48, 50, 38, 40 | add4d 8212 |
. . . . . . . . . . . . . 14
|
| 52 | 19, 43 | oveq12d 5943 |
. . . . . . . . . . . . . 14
|
| 53 | 35, 49, 39 | adddid 8068 |
. . . . . . . . . . . . . . 15
|
| 54 | 53 | oveq2d 5941 |
. . . . . . . . . . . . . 14
|
| 55 | 51, 52, 54 | 3eqtr4d 2239 |
. . . . . . . . . . . . 13
|
| 56 | 47, 55 | breq12d 4047 |
. . . . . . . . . . . 12
#
# |
| 57 | reapadd1 8640 |
. . . . . . . . . . . . . 14
#
# | |
| 58 | 10, 13, 23, 57 | syl3anc 1249 |
. . . . . . . . . . . . 13
#
# |
| 59 | reapadd1 8640 |
. . . . . . . . . . . . . 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 4034
(class class class)co 5925 cc 7894 cr 7895 ci 7898 caddc 7899 cmul 7901 # cap 8625 |
| 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 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 |
| 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 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-br 4035 df-opab 4096 df-id 4329 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-iota 5220 df-fun 5261 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-pnf 8080 df-mnf 8081 df-ltxr 8083 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 |
| This theorem is referenced by: apadd2 8653 addext 8654 apsub1 8686 subap0 8687 |
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