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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 7661 |
. . . 4
 | |
| 2 | divadddivap 8168 |
. . . 4
#
#  | |
| 3 | 1, 2 | sylanl2 395 |
. . 3
#
# |
| 4 | simplr 497 |
. . . . . 6
#
# | |
| 5 | simprrl 506 |
. . . . . 6
#
# | |
| 6 | simprrr 507 |
. . . . . 6
#
# # | |
| 7 | divnegap 8147 |
. . . . . 6
# | |
| 8 | 4, 5, 6, 7 | syl3anc 1174 |
. . . . 5
#
# |
| 9 | 8 | oveq2d 5650 |
. . . 4
#
# |
| 10 | simpll 496 |
. . . . . 6
#
# | |
| 11 | simprll 504 |
. . . . . 6
#
# | |
| 12 | simprlr 505 |
. . . . . 6
#
# # | |
| 13 | divclap 8119 |
. . . . . 6
# | |
| 14 | 10, 11, 12, 13 | syl3anc 1174 |
. . . . 5
#
# |
| 15 | divclap 8119 |
. . . . . 6
# | |
| 16 | 4, 5, 6, 15 | syl3anc 1174 |
. . . . 5
#
# |
| 17 | 14, 16 | negsubd 7778 |
. . . 4
#
# |
| 18 | 9, 17 | eqtr3d 2122 |
. . 3
#
# |
| 19 | 3, 18 | eqtr3d 2122 |
. 2
#
# |
| 20 | 4, 11 | mulneg1d 7868 |
. . . . 5
#
#
|
| 21 | 20 | oveq2d 5650 |
. . . 4
#
#
|
| 22 | 10, 5 | mulcld 7487 |
. . . . 5
#
# |
| 23 | 4, 11 | mulcld 7487 |
. . . . 5
#
# |
| 24 | 22, 23 | negsubd 7778 |
. . . 4
#
#
|
| 25 | 21, 24 | eqtrd 2120 |
. . 3
#
#
|
| 26 | 25 | oveq1d 5649 |
. 2
#
#
|
| 27 | 19, 26 | eqtr3d 2122 |
1
#
#
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 102
wceq 1289
wcel 1438
class class class wbr 3837 (class class class)co 5634
cc 7327
cc0 7329
caddc 7332 cmul 7334 cmin 7632 cneg 7633 # cap 8034
cdiv 8113 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3949 ax-pow 4001 ax-pr 4027 ax-un 4251 ax-setind 4343 ax-cnex 7415 ax-resscn 7416 ax-1cn 7417 ax-1re 7418 ax-icn 7419 ax-addcl 7420 ax-addrcl 7421 ax-mulcl 7422 ax-mulrcl 7423 ax-addcom 7424 ax-mulcom 7425 ax-addass 7426 ax-mulass 7427 ax-distr 7428 ax-i2m1 7429 ax-0lt1 7430 ax-1rid 7431 ax-0id 7432 ax-rnegex 7433 ax-precex 7434 ax-cnre 7435 ax-pre-ltirr 7436 ax-pre-ltwlin 7437 ax-pre-lttrn 7438 ax-pre-apti 7439 ax-pre-ltadd 7440 ax-pre-mulgt0 7441 ax-pre-mulext 7442 |
| This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rmo 2367 df-rab 2368 df-v 2621 df-sbc 2839 df-dif 2999 df-un 3001 df-in 3003 df-ss 3010 df-pw 3427 df-sn 3447 df-pr 3448 df-op 3450 df-uni 3649 df-br 3838 df-opab 3892 df-id 4111 df-po 4114 df-iso 4115 df-xp 4434 df-rel 4435 df-cnv 4436 df-co 4437 df-dm 4438 df-iota 4967 df-fun 5004 df-fv 5010 df-riota 5590 df-ov 5637 df-oprab 5638 df-mpt2 5639 df-pnf 7503 df-mnf 7504 df-xr 7505 df-ltxr 7506 df-le 7507 df-sub 7634 df-neg 7635 df-reap 8028 df-ap 8035 df-div 8114 |
| This theorem is referenced by: (None) |
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