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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 8157 |
. . . 4
 | |
| 2 | divadddivap 8684 |
. . . 4
#
#  | |
| 3 | 1, 2 | sylanl2 403 |
. . 3
#
# |
| 4 | simplr 528 |
. . . . . 6
#
# | |
| 5 | simprrl 539 |
. . . . . 6
#
# | |
| 6 | simprrr 540 |
. . . . . 6
#
# # | |
| 7 | divnegap 8663 |
. . . . . 6
# | |
| 8 | 4, 5, 6, 7 | syl3anc 1238 |
. . . . 5
#
# |
| 9 | 8 | oveq2d 5891 |
. . . 4
#
# |
| 10 | simpll 527 |
. . . . . 6
#
# | |
| 11 | simprll 537 |
. . . . . 6
#
# | |
| 12 | simprlr 538 |
. . . . . 6
#
# # | |
| 13 | divclap 8635 |
. . . . . 6
# | |
| 14 | 10, 11, 12, 13 | syl3anc 1238 |
. . . . 5
#
# |
| 15 | divclap 8635 |
. . . . . 6
# | |
| 16 | 4, 5, 6, 15 | syl3anc 1238 |
. . . . 5
#
# |
| 17 | 14, 16 | negsubd 8274 |
. . . 4
#
# |
| 18 | 9, 17 | eqtr3d 2212 |
. . 3
#
# |
| 19 | 3, 18 | eqtr3d 2212 |
. 2
#
# |
| 20 | 4, 11 | mulneg1d 8368 |
. . . . 5
#
#
|
| 21 | 20 | oveq2d 5891 |
. . . 4
#
#
|
| 22 | 10, 5 | mulcld 7978 |
. . . . 5
#
# |
| 23 | 4, 11 | mulcld 7978 |
. . . . 5
#
# |
| 24 | 22, 23 | negsubd 8274 |
. . . 4
#
#
|
| 25 | 21, 24 | eqtrd 2210 |
. . 3
#
#
|
| 26 | 25 | oveq1d 5890 |
. 2
#
#
|
| 27 | 19, 26 | eqtr3d 2212 |
1
#
#
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wceq 1353
wcel 2148
class class class wbr 4004 (class class class)co 5875
cc 7809
cc0 7811
caddc 7814 cmul 7816 cmin 8128 cneg 8129 # cap 8538
cdiv 8629 |
| 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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 ax-pre-mulext 7929 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2740 df-sbc 2964 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-br 4005 df-opab 4066 df-id 4294 df-po 4297 df-iso 4298 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-iota 5179 df-fun 5219 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539 df-div 8630 |
| This theorem is referenced by: subrecap 8796 |
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