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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 7743 |
. . . 4
 | |
| 2 | divadddivap 8255 |
. . . 4
#
#  | |
| 3 | 1, 2 | sylanl2 396 |
. . 3
#
# |
| 4 | simplr 498 |
. . . . . 6
#
# | |
| 5 | simprrl 507 |
. . . . . 6
#
# | |
| 6 | simprrr 508 |
. . . . . 6
#
# # | |
| 7 | divnegap 8234 |
. . . . . 6
# | |
| 8 | 4, 5, 6, 7 | syl3anc 1175 |
. . . . 5
#
# |
| 9 | 8 | oveq2d 5682 |
. . . 4
#
# |
| 10 | simpll 497 |
. . . . . 6
#
# | |
| 11 | simprll 505 |
. . . . . 6
#
# | |
| 12 | simprlr 506 |
. . . . . 6
#
# # | |
| 13 | divclap 8206 |
. . . . . 6
# | |
| 14 | 10, 11, 12, 13 | syl3anc 1175 |
. . . . 5
#
# |
| 15 | divclap 8206 |
. . . . . 6
# | |
| 16 | 4, 5, 6, 15 | syl3anc 1175 |
. . . . 5
#
# |
| 17 | 14, 16 | negsubd 7860 |
. . . 4
#
# |
| 18 | 9, 17 | eqtr3d 2123 |
. . 3
#
# |
| 19 | 3, 18 | eqtr3d 2123 |
. 2
#
# |
| 20 | 4, 11 | mulneg1d 7950 |
. . . . 5
#
#
|
| 21 | 20 | oveq2d 5682 |
. . . 4
#
#
|
| 22 | 10, 5 | mulcld 7569 |
. . . . 5
#
# |
| 23 | 4, 11 | mulcld 7569 |
. . . . 5
#
# |
| 24 | 22, 23 | negsubd 7860 |
. . . 4
#
#
|
| 25 | 21, 24 | eqtrd 2121 |
. . 3
#
#
|
| 26 | 25 | oveq1d 5681 |
. 2
#
#
|
| 27 | 19, 26 | eqtr3d 2123 |
1
#
#
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wceq 1290
wcel 1439
class class class wbr 3851 (class class class)co 5666
cc 7409
cc0 7411
caddc 7414 cmul 7416 cmin 7714 cneg 7715 # cap 8119
cdiv 8200 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-mulrcl 7505 ax-addcom 7506 ax-mulcom 7507 ax-addass 7508 ax-mulass 7509 ax-distr 7510 ax-i2m1 7511 ax-0lt1 7512 ax-1rid 7513 ax-0id 7514 ax-rnegex 7515 ax-precex 7516 ax-cnre 7517 ax-pre-ltirr 7518 ax-pre-ltwlin 7519 ax-pre-lttrn 7520 ax-pre-apti 7521 ax-pre-ltadd 7522 ax-pre-mulgt0 7523 ax-pre-mulext 7524 |
| This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-id 4129 df-po 4132 df-iso 4133 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-pnf 7585 df-mnf 7586 df-xr 7587 df-ltxr 7588 df-le 7589 df-sub 7716 df-neg 7717 df-reap 8113 df-ap 8120 df-div 8201 |
| This theorem is referenced by: (None) |
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