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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 8254 |
. . . 4
 | |
| 2 | divadddivap 8782 |
. . . 4
#
#  | |
| 3 | 1, 2 | sylanl2 403 |
. . 3
#
# |
| 4 | simplr 528 |
. . . . . 6
#
# | |
| 5 | simprrl 539 |
. . . . . 6
#
# | |
| 6 | simprrr 540 |
. . . . . 6
#
# # | |
| 7 | divnegap 8761 |
. . . . . 6
# | |
| 8 | 4, 5, 6, 7 | syl3anc 1249 |
. . . . 5
#
# |
| 9 | 8 | oveq2d 5950 |
. . . 4
#
# |
| 10 | simpll 527 |
. . . . . 6
#
# | |
| 11 | simprll 537 |
. . . . . 6
#
# | |
| 12 | simprlr 538 |
. . . . . 6
#
# # | |
| 13 | divclap 8733 |
. . . . . 6
# | |
| 14 | 10, 11, 12, 13 | syl3anc 1249 |
. . . . 5
#
# |
| 15 | divclap 8733 |
. . . . . 6
# | |
| 16 | 4, 5, 6, 15 | syl3anc 1249 |
. . . . 5
#
# |
| 17 | 14, 16 | negsubd 8371 |
. . . 4
#
# |
| 18 | 9, 17 | eqtr3d 2239 |
. . 3
#
# |
| 19 | 3, 18 | eqtr3d 2239 |
. 2
#
# |
| 20 | 4, 11 | mulneg1d 8465 |
. . . . 5
#
#
|
| 21 | 20 | oveq2d 5950 |
. . . 4
#
#
|
| 22 | 10, 5 | mulcld 8075 |
. . . . 5
#
# |
| 23 | 4, 11 | mulcld 8075 |
. . . . 5
#
# |
| 24 | 22, 23 | negsubd 8371 |
. . . 4
#
#
|
| 25 | 21, 24 | eqtrd 2237 |
. . 3
#
#
|
| 26 | 25 | oveq1d 5949 |
. 2
#
#
|
| 27 | 19, 26 | eqtr3d 2239 |
1
#
#
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wceq 1372
wcel 2175
class class class wbr 4043 (class class class)co 5934
cc 7905
cc0 7907
caddc 7910 cmul 7912 cmin 8225 cneg 8226 # cap 8636
cdiv 8727 |
| 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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4478 ax-setind 4583 ax-cnex 7998 ax-resscn 7999 ax-1cn 8000 ax-1re 8001 ax-icn 8002 ax-addcl 8003 ax-addrcl 8004 ax-mulcl 8005 ax-mulrcl 8006 ax-addcom 8007 ax-mulcom 8008 ax-addass 8009 ax-mulass 8010 ax-distr 8011 ax-i2m1 8012 ax-0lt1 8013 ax-1rid 8014 ax-0id 8015 ax-rnegex 8016 ax-precex 8017 ax-cnre 8018 ax-pre-ltirr 8019 ax-pre-ltwlin 8020 ax-pre-lttrn 8021 ax-pre-apti 8022 ax-pre-ltadd 8023 ax-pre-mulgt0 8024 ax-pre-mulext 8025 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rmo 2491 df-rab 2492 df-v 2773 df-sbc 2998 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-br 4044 df-opab 4105 df-id 4338 df-po 4341 df-iso 4342 df-xp 4679 df-rel 4680 df-cnv 4681 df-co 4682 df-dm 4683 df-iota 5229 df-fun 5270 df-fv 5276 df-riota 5889 df-ov 5937 df-oprab 5938 df-mpo 5939 df-pnf 8091 df-mnf 8092 df-xr 8093 df-ltxr 8094 df-le 8095 df-sub 8227 df-neg 8228 df-reap 8630 df-ap 8637 df-div 8728 |
| This theorem is referenced by: subrecap 8894 |
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