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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 7955 |
. . . 4
 | |
| 2 | divadddivap 8480 |
. . . 4
#
#  | |
| 3 | 1, 2 | sylanl2 400 |
. . 3
#
# |
| 4 | simplr 519 |
. . . . . 6
#
# | |
| 5 | simprrl 528 |
. . . . . 6
#
# | |
| 6 | simprrr 529 |
. . . . . 6
#
# # | |
| 7 | divnegap 8459 |
. . . . . 6
# | |
| 8 | 4, 5, 6, 7 | syl3anc 1216 |
. . . . 5
#
# |
| 9 | 8 | oveq2d 5783 |
. . . 4
#
# |
| 10 | simpll 518 |
. . . . . 6
#
# | |
| 11 | simprll 526 |
. . . . . 6
#
# | |
| 12 | simprlr 527 |
. . . . . 6
#
# # | |
| 13 | divclap 8431 |
. . . . . 6
# | |
| 14 | 10, 11, 12, 13 | syl3anc 1216 |
. . . . 5
#
# |
| 15 | divclap 8431 |
. . . . . 6
# | |
| 16 | 4, 5, 6, 15 | syl3anc 1216 |
. . . . 5
#
# |
| 17 | 14, 16 | negsubd 8072 |
. . . 4
#
# |
| 18 | 9, 17 | eqtr3d 2172 |
. . 3
#
# |
| 19 | 3, 18 | eqtr3d 2172 |
. 2
#
# |
| 20 | 4, 11 | mulneg1d 8166 |
. . . . 5
#
#
|
| 21 | 20 | oveq2d 5783 |
. . . 4
#
#
|
| 22 | 10, 5 | mulcld 7779 |
. . . . 5
#
# |
| 23 | 4, 11 | mulcld 7779 |
. . . . 5
#
# |
| 24 | 22, 23 | negsubd 8072 |
. . . 4
#
#
|
| 25 | 21, 24 | eqtrd 2170 |
. . 3
#
#
|
| 26 | 25 | oveq1d 5782 |
. 2
#
#
|
| 27 | 19, 26 | eqtr3d 2172 |
1
#
#
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wceq 1331
wcel 1480
class class class wbr 3924 (class class class)co 5767
cc 7611
cc0 7613
caddc 7616 cmul 7618 cmin 7926 cneg 7927 # cap 8336
cdiv 8425 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 |
| This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 |
| This theorem is referenced by: subrecap 8591 |
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