Nearby theorems
|
|
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
| 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 8292 |
. . . 4
 | |
| 2 | divadddivap 8820 |
. . . 4
#
#  | |
| 3 | 1, 2 | sylanl2 403 |
. . 3
#
# |
| 4 | simplr 528 |
. . . . . 6
#
# | |
| 5 | simprrl 539 |
. . . . . 6
#
# | |
| 6 | simprrr 540 |
. . . . . 6
#
# # | |
| 7 | divnegap 8799 |
. . . . . 6
# | |
| 8 | 4, 5, 6, 7 | syl3anc 1250 |
. . . . 5
#
# |
| 9 | 8 | oveq2d 5973 |
. . . 4
#
# |
| 10 | simpll 527 |
. . . . . 6
#
# | |
| 11 | simprll 537 |
. . . . . 6
#
# | |
| 12 | simprlr 538 |
. . . . . 6
#
# # | |
| 13 | divclap 8771 |
. . . . . 6
# | |
| 14 | 10, 11, 12, 13 | syl3anc 1250 |
. . . . 5
#
# |
| 15 | divclap 8771 |
. . . . . 6
# | |
| 16 | 4, 5, 6, 15 | syl3anc 1250 |
. . . . 5
#
# |
| 17 | 14, 16 | negsubd 8409 |
. . . 4
#
# |
| 18 | 9, 17 | eqtr3d 2241 |
. . 3
#
# |
| 19 | 3, 18 | eqtr3d 2241 |
. 2
#
# |
| 20 | 4, 11 | mulneg1d 8503 |
. . . . 5
#
#
|
| 21 | 20 | oveq2d 5973 |
. . . 4
#
#
|
| 22 | 10, 5 | mulcld 8113 |
. . . . 5
#
# |
| 23 | 4, 11 | mulcld 8113 |
. . . . 5
#
# |
| 24 | 22, 23 | negsubd 8409 |
. . . 4
#
#
|
| 25 | 21, 24 | eqtrd 2239 |
. . 3
#
#
|
| 26 | 25 | oveq1d 5972 |
. 2
#
#
|
| 27 | 19, 26 | eqtr3d 2241 |
1
#
#
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wceq 1373
wcel 2177
class class class wbr 4051 (class class class)co 5957
cc 7943
cc0 7945
caddc 7948 cmul 7950 cmin 8263 cneg 8264 # cap 8674
cdiv 8765 |
| 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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-sep 4170 ax-pow 4226 ax-pr 4261 ax-un 4488 ax-setind 4593 ax-cnex 8036 ax-resscn 8037 ax-1cn 8038 ax-1re 8039 ax-icn 8040 ax-addcl 8041 ax-addrcl 8042 ax-mulcl 8043 ax-mulrcl 8044 ax-addcom 8045 ax-mulcom 8046 ax-addass 8047 ax-mulass 8048 ax-distr 8049 ax-i2m1 8050 ax-0lt1 8051 ax-1rid 8052 ax-0id 8053 ax-rnegex 8054 ax-precex 8055 ax-cnre 8056 ax-pre-ltirr 8057 ax-pre-ltwlin 8058 ax-pre-lttrn 8059 ax-pre-apti 8060 ax-pre-ltadd 8061 ax-pre-mulgt0 8062 ax-pre-mulext 8063 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3003 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3857 df-br 4052 df-opab 4114 df-id 4348 df-po 4351 df-iso 4352 df-xp 4689 df-rel 4690 df-cnv 4691 df-co 4692 df-dm 4693 df-iota 5241 df-fun 5282 df-fv 5288 df-riota 5912 df-ov 5960 df-oprab 5961 df-mpo 5962 df-pnf 8129 df-mnf 8130 df-xr 8131 df-ltxr 8132 df-le 8133 df-sub 8265 df-neg 8266 df-reap 8668 df-ap 8675 df-div 8766 |
| This theorem is referenced by: subrecap 8932 |
| Copyright terms: Public domain | W3C validator |