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| Mirrors > Home > ILE Home > Th. List > apadd1 | Unicode version | ||
| Description: Addition respects apartness. Analogue of addcan 8358 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 8174 |
. . 3
     | |
| 2 | 1 | 3ad2ant3 1046 |
. 2
|
| 3 | cnre 8174 |
. . . . . . 7
| |
| 4 | 3 | 3ad2ant2 1045 |
. . . . . 6
|
| 5 | 4 | ad2antrr 488 |
. . . . 5
|
| 6 | cnre 8174 |
. . . . . . . . . . 11
| |
| 7 | 6 | 3ad2ant1 1044 |
. . . . . . . . . 10
|
| 8 | 7 | ad2antrr 488 |
. . . . . . . . 9
|
| 9 | 8 | ad2antrr 488 |
. . . . . . . 8
|
| 10 | simplrl 537 |
. . . . . . . . . . . 12
| |
| 11 | simplrr 538 |
. . . . . . . . . . . 12
| |
| 12 | simprl 531 |
. . . . . . . . . . . . 13
| |
| 13 | 12 | ad3antrrr 492 |
. . . . . . . . . . . 12
|
| 14 | simprr 533 |
. . . . . . . . . . . . 13
| |
| 15 | 14 | ad3antrrr 492 |
. . . . . . . . . . . 12
|
| 16 | apreim 8782 |
. . . . . . . . . . . 12
#
# 
# | |
| 17 | 10, 11, 13, 15, 16 | syl22anc 1274 |
. . . . . . . . . . 11
#
#
# |
| 18 | simpr 110 |
. . . . . . . . . . . 12
| |
| 19 | simpllr 536 |
. . . . . . . . . . . 12
| |
| 20 | 18, 19 | breq12d 4101 |
. . . . . . . . . . 11
#
# |
| 21 | simprl 531 |
. . . . . . . . . . . . . . . 16
| |
| 22 | 21 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
|
| 23 | 22 | ad3antrrr 492 |
. . . . . . . . . . . . . 14
|
| 24 | 10, 23 | readdcld 8208 |
. . . . . . . . . . . . 13
|
| 25 | simprr 533 |
. . . . . . . . . . . . . . . 16
| |
| 26 | 25 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
|
| 27 | 26 | ad3antrrr 492 |
. . . . . . . . . . . . . 14
|
| 28 | 11, 27 | readdcld 8208 |
. . . . . . . . . . . . 13
|
| 29 | 13, 23 | readdcld 8208 |
. . . . . . . . . . . . 13
|
| 30 | 15, 27 | readdcld 8208 |
. . . . . . . . . . . . 13
|
| 31 | apreim 8782 |
. . . . . . . . . . . . 13
#
# # | |
| 32 | 24, 28, 29, 30, 31 | syl22anc 1274 |
. . . . . . . . . . . 12
#
# # |
| 33 | 10 | recnd 8207 |
. . . . . . . . . . . . . . 15
|
| 34 | ax-icn 8126 |
. . . . . . . . . . . . . . . . 17
| |
| 35 | 34 | a1i 9 |
. . . . . . . . . . . . . . . 16
|
| 36 | 11 | recnd 8207 |
. . . . . . . . . . . . . . . 16
|
| 37 | 35, 36 | mulcld 8199 |
. . . . . . . . . . . . . . 15
|
| 38 | 23 | recnd 8207 |
. . . . . . . . . . . . . . 15
|
| 39 | 27 | recnd 8207 |
. . . . . . . . . . . . . . . 16
|
| 40 | 35, 39 | mulcld 8199 |
. . . . . . . . . . . . . . 15
|
| 41 | 33, 37, 38, 40 | add4d 8347 |
. . . . . . . . . . . . . 14
|
| 42 | simplr 529 |
. . . . . . . . . . . . . . . 16
| |
| 43 | 42 | ad3antrrr 492 |
. . . . . . . . . . . . . . 15
|
| 44 | 18, 43 | oveq12d 6035 |
. . . . . . . . . . . . . 14
|
| 45 | 35, 36, 39 | adddid 8203 |
. . . . . . . . . . . . . . 15
|
| 46 | 45 | oveq2d 6033 |
. . . . . . . . . . . . . 14
|
| 47 | 41, 44, 46 | 3eqtr4d 2274 |
. . . . . . . . . . . . 13
|
| 48 | 13 | recnd 8207 |
. . . . . . . . . . . . . . 15
|
| 49 | 15 | recnd 8207 |
. . . . . . . . . . . . . . . 16
|
| 50 | 35, 49 | mulcld 8199 |
. . . . . . . . . . . . . . 15
|
| 51 | 48, 50, 38, 40 | add4d 8347 |
. . . . . . . . . . . . . 14
|
| 52 | 19, 43 | oveq12d 6035 |
. . . . . . . . . . . . . 14
|
| 53 | 35, 49, 39 | adddid 8203 |
. . . . . . . . . . . . . . 15
|
| 54 | 53 | oveq2d 6033 |
. . . . . . . . . . . . . 14
|
| 55 | 51, 52, 54 | 3eqtr4d 2274 |
. . . . . . . . . . . . 13
|
| 56 | 47, 55 | breq12d 4101 |
. . . . . . . . . . . 12
#
# |
| 57 | reapadd1 8775 |
. . . . . . . . . . . . . 14
#
# | |
| 58 | 10, 13, 23, 57 | syl3anc 1273 |
. . . . . . . . . . . . 13
#
# |
| 59 | reapadd1 8775 |
. . . . . . . . . . . . . 14
#
# | |
| 60 | 11, 15, 27, 59 | syl3anc 1273 |
. . . . . . . . . . . . 13
#
# |
| 61 | 58, 60 | orbi12d 800 |
. . . . . . . . . . . 12
# #
# # |
| 62 | 32, 56, 61 | 3bitr4d 220 |
. . . . . . . . . . 11
#
#
# |
| 63 | 17, 20, 62 | 3bitr4d 220 |
. . . . . . . . . 10
#
#
|
| 64 | 63 | ex 115 |
. . . . . . . . 9
#
#
|
| 65 | 64 | rexlimdvva 2658 |
. . . . . . . 8
#
#
|
| 66 | 9, 65 | mpd 13 |
. . . . . . 7
#
#
|
| 67 | 66 | ex 115 |
. . . . . 6
#
#
|
| 68 | 67 | rexlimdvva 2658 |
. . . . 5
#
#
|
| 69 | 5, 68 | mpd 13 |
. . . 4
#
#
|
| 70 | 69 | ex 115 |
. . 3
#
#
|
| 71 | 70 | rexlimdvva 2658 |
. 2
#
#
|
| 72 | 2, 71 | mpd 13 |
1
#
#
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wo 715 w3a 1004 wceq 1397 wcel 2202 wrex 2511 class class class wbr 4088
(class class class)co 6017 cc 8029 cr 8030 ci 8033 caddc 8034 cmul 8036 # cap 8760 |
| 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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148 |
| This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-opab 4151 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-iota 5286 df-fun 5328 df-fv 5334 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-pnf 8215 df-mnf 8216 df-ltxr 8218 df-sub 8351 df-neg 8352 df-reap 8754 df-ap 8761 |
| This theorem is referenced by: apadd2 8788 addext 8789 apsub1 8821 subap0 8822 |
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