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| Mirrors > Home > ILE Home > Th. List > divsubdivap | Unicode version | ||
| Description: Subtraction of two ratios. (Contributed by Jim Kingdon, 26-Feb-2020.) |
| Ref | Expression |
|---|---|
| divsubdivap |
     ![/\](wedge.gif "/\"){kind=small}    #  
#   
 
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | negcl 8369 |
. . . 4
 | |
| 2 | divadddivap 8897 |
. . . 4
#
#  | |
| 3 | 1, 2 | sylanl2 403 |
. . 3
#
# |
| 4 | simplr 528 |
. . . . . 6
#
# | |
| 5 | simprrl 539 |
. . . . . 6
#
# | |
| 6 | simprrr 540 |
. . . . . 6
#
# # | |
| 7 | divnegap 8876 |
. . . . . 6
# | |
| 8 | 4, 5, 6, 7 | syl3anc 1271 |
. . . . 5
#
# |
| 9 | 8 | oveq2d 6029 |
. . . 4
#
# |
| 10 | simpll 527 |
. . . . . 6
#
# | |
| 11 | simprll 537 |
. . . . . 6
#
# | |
| 12 | simprlr 538 |
. . . . . 6
#
# # | |
| 13 | divclap 8848 |
. . . . . 6
# | |
| 14 | 10, 11, 12, 13 | syl3anc 1271 |
. . . . 5
#
# |
| 15 | divclap 8848 |
. . . . . 6
# | |
| 16 | 4, 5, 6, 15 | syl3anc 1271 |
. . . . 5
#
# |
| 17 | 14, 16 | negsubd 8486 |
. . . 4
#
# |
| 18 | 9, 17 | eqtr3d 2264 |
. . 3
#
# |
| 19 | 3, 18 | eqtr3d 2264 |
. 2
#
# |
| 20 | 4, 11 | mulneg1d 8580 |
. . . . 5
#
#
|
| 21 | 20 | oveq2d 6029 |
. . . 4
#
#
|
| 22 | 10, 5 | mulcld 8190 |
. . . . 5
#
# |
| 23 | 4, 11 | mulcld 8190 |
. . . . 5
#
# |
| 24 | 22, 23 | negsubd 8486 |
. . . 4
#
#
|
| 25 | 21, 24 | eqtrd 2262 |
. . 3
#
#
|
| 26 | 25 | oveq1d 6028 |
. 2
#
#
|
| 27 | 19, 26 | eqtr3d 2264 |
1
#
#
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wceq 1395
wcel 2200
class class class wbr 4086 (class class class)co 6013
cc 8020
cc0 8022
caddc 8025 cmul 8027 cmin 8340 cneg 8341 # cap 8751
cdiv 8842 |
| 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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4205 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-cnex 8113 ax-resscn 8114 ax-1cn 8115 ax-1re 8116 ax-icn 8117 ax-addcl 8118 ax-addrcl 8119 ax-mulcl 8120 ax-mulrcl 8121 ax-addcom 8122 ax-mulcom 8123 ax-addass 8124 ax-mulass 8125 ax-distr 8126 ax-i2m1 8127 ax-0lt1 8128 ax-1rid 8129 ax-0id 8130 ax-rnegex 8131 ax-precex 8132 ax-cnre 8133 ax-pre-ltirr 8134 ax-pre-ltwlin 8135 ax-pre-lttrn 8136 ax-pre-apti 8137 ax-pre-ltadd 8138 ax-pre-mulgt0 8139 ax-pre-mulext 8140 |
| This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2802 df-sbc 3030 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-br 4087 df-opab 4149 df-id 4388 df-po 4391 df-iso 4392 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-iota 5284 df-fun 5326 df-fv 5332 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-pnf 8206 df-mnf 8207 df-xr 8208 df-ltxr 8209 df-le 8210 df-sub 8342 df-neg 8343 df-reap 8745 df-ap 8752 df-div 8843 |
| This theorem is referenced by: subrecap 9009 |
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