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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 8188 |
. . . 4
 | |
| 2 | divadddivap 8715 |
. . . 4
#
#  | |
| 3 | 1, 2 | sylanl2 403 |
. . 3
#
# |
| 4 | simplr 528 |
. . . . . 6
#
# | |
| 5 | simprrl 539 |
. . . . . 6
#
# | |
| 6 | simprrr 540 |
. . . . . 6
#
# # | |
| 7 | divnegap 8694 |
. . . . . 6
# | |
| 8 | 4, 5, 6, 7 | syl3anc 1249 |
. . . . 5
#
# |
| 9 | 8 | oveq2d 5913 |
. . . 4
#
# |
| 10 | simpll 527 |
. . . . . 6
#
# | |
| 11 | simprll 537 |
. . . . . 6
#
# | |
| 12 | simprlr 538 |
. . . . . 6
#
# # | |
| 13 | divclap 8666 |
. . . . . 6
# | |
| 14 | 10, 11, 12, 13 | syl3anc 1249 |
. . . . 5
#
# |
| 15 | divclap 8666 |
. . . . . 6
# | |
| 16 | 4, 5, 6, 15 | syl3anc 1249 |
. . . . 5
#
# |
| 17 | 14, 16 | negsubd 8305 |
. . . 4
#
# |
| 18 | 9, 17 | eqtr3d 2224 |
. . 3
#
# |
| 19 | 3, 18 | eqtr3d 2224 |
. 2
#
# |
| 20 | 4, 11 | mulneg1d 8399 |
. . . . 5
#
#
|
| 21 | 20 | oveq2d 5913 |
. . . 4
#
#
|
| 22 | 10, 5 | mulcld 8009 |
. . . . 5
#
# |
| 23 | 4, 11 | mulcld 8009 |
. . . . 5
#
# |
| 24 | 22, 23 | negsubd 8305 |
. . . 4
#
#
|
| 25 | 21, 24 | eqtrd 2222 |
. . 3
#
#
|
| 26 | 25 | oveq1d 5912 |
. 2
#
#
|
| 27 | 19, 26 | eqtr3d 2224 |
1
#
#
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wceq 1364
wcel 2160
class class class wbr 4018 (class class class)co 5897
cc 7840
cc0 7842
caddc 7845 cmul 7847 cmin 8159 cneg 8160 # cap 8569
cdiv 8660 |
| 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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-div 8661 |
| This theorem is referenced by: subrecap 8827 |
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