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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 8379 |
. . . 4
 | |
| 2 | divadddivap 8907 |
. . . 4
#
#  | |
| 3 | 1, 2 | sylanl2 403 |
. . 3
#
# |
| 4 | simplr 529 |
. . . . . 6
#
# | |
| 5 | simprrl 541 |
. . . . . 6
#
# | |
| 6 | simprrr 542 |
. . . . . 6
#
# # | |
| 7 | divnegap 8886 |
. . . . . 6
# | |
| 8 | 4, 5, 6, 7 | syl3anc 1273 |
. . . . 5
#
# |
| 9 | 8 | oveq2d 6034 |
. . . 4
#
# |
| 10 | simpll 527 |
. . . . . 6
#
# | |
| 11 | simprll 539 |
. . . . . 6
#
# | |
| 12 | simprlr 540 |
. . . . . 6
#
# # | |
| 13 | divclap 8858 |
. . . . . 6
# | |
| 14 | 10, 11, 12, 13 | syl3anc 1273 |
. . . . 5
#
# |
| 15 | divclap 8858 |
. . . . . 6
# | |
| 16 | 4, 5, 6, 15 | syl3anc 1273 |
. . . . 5
#
# |
| 17 | 14, 16 | negsubd 8496 |
. . . 4
#
# |
| 18 | 9, 17 | eqtr3d 2266 |
. . 3
#
# |
| 19 | 3, 18 | eqtr3d 2266 |
. 2
#
# |
| 20 | 4, 11 | mulneg1d 8590 |
. . . . 5
#
#
|
| 21 | 20 | oveq2d 6034 |
. . . 4
#
#
|
| 22 | 10, 5 | mulcld 8200 |
. . . . 5
#
# |
| 23 | 4, 11 | mulcld 8200 |
. . . . 5
#
# |
| 24 | 22, 23 | negsubd 8496 |
. . . 4
#
#
|
| 25 | 21, 24 | eqtrd 2264 |
. . 3
#
#
|
| 26 | 25 | oveq1d 6033 |
. 2
#
#
|
| 27 | 19, 26 | eqtr3d 2266 |
1
#
#
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wceq 1397
wcel 2202
class class class wbr 4088 (class class class)co 6018
cc 8030
cc0 8032
caddc 8035 cmul 8037 cmin 8350 cneg 8351 # cap 8761
cdiv 8852 |
| 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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 |
| This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-opab 4151 df-id 4390 df-po 4393 df-iso 4394 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-iota 5286 df-fun 5328 df-fv 5334 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 |
| This theorem is referenced by: subrecap 9019 |
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