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| Mirrors > Home > ILE Home > Th. List > divdivdivap | Unicode version | ||
| Description: Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by Jim Kingdon, 25-Feb-2020.) |
| Ref | Expression |
|---|---|
| divdivdivap |
     ![/\](wedge.gif "/\"){kind=small}  #    # 
#     |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simprrl 539 |
. . . . . . 7
# #
# | |
| 2 | simprll 537 |
. . . . . . 7
# #
# | |
| 3 | simprlr 538 |
. . . . . . 7
# #
# # | |
| 4 | divclap 8654 |
. . . . . . 7
# | |
| 5 | 1, 2, 3, 4 | syl3anc 1249 |
. . . . . 6
# #
# |
| 6 | simpll 527 |
. . . . . . 7
# #
# | |
| 7 | simplrl 535 |
. . . . . . 7
# #
# | |
| 8 | simplrr 536 |
. . . . . . 7
# #
# # | |
| 9 | divclap 8654 |
. . . . . . 7
# | |
| 10 | 6, 7, 8, 9 | syl3anc 1249 |
. . . . . 6
# #
# |
| 11 | 5, 10 | mulcomd 7998 |
. . . . 5
# #
# |
| 12 | simplr 528 |
. . . . . 6
# #
# # | |
| 13 | simprl 529 |
. . . . . 6
# #
# # | |
| 14 | divmuldivap 8688 |
. . . . . 6
#
# | |
| 15 | 6, 1, 12, 13, 14 | syl22anc 1250 |
. . . . 5
# #
# |
| 16 | 11, 15 | eqtrd 2222 |
. . . 4
# #
# |
| 17 | 16 | oveq2d 5907 |
. . 3
# #
# |
| 18 | simprr 531 |
. . . . . . 7
# #
# # | |
| 19 | divmuldivap 8688 |
. . . . . . 7
#
# | |
| 20 | 2, 1, 18, 13, 19 | syl22anc 1250 |
. . . . . 6
# #
# |
| 21 | 2, 1 | mulcomd 7998 |
. . . . . . . 8
# #
# |
| 22 | 21 | oveq1d 5906 |
. . . . . . 7
# #
# |
| 23 | 1, 2 | mulcld 7997 |
. . . . . . . 8
# #
# |
| 24 | simprrr 540 |
. . . . . . . . 9
# #
# # | |
| 25 | 1, 2, 24, 3 | mulap0d 8634 |
. . . . . . . 8
# #
# #
|
| 26 | dividap 8677 |
. . . . . . . 8
#
 | |
| 27 | 23, 25, 26 | syl2anc 411 |
. . . . . . 7
# #
# |
| 28 | 22, 27 | eqtrd 2222 |
. . . . . 6
# #
# |
| 29 | 20, 28 | eqtrd 2222 |
. . . . 5
# #
# |
| 30 | 29 | oveq1d 5906 |
. . . 4
# #
# |
| 31 | divclap 8654 |
. . . . . 6
# | |
| 32 | 2, 1, 24, 31 | syl3anc 1249 |
. . . . 5
# #
# |
| 33 | 32, 5, 10 | mulassd 8000 |
. . . 4
# #
# |
| 34 | 10 | mulid2d 7995 |
. . . 4
# #
# |
| 35 | 30, 33, 34 | 3eqtr3d 2230 |
. . 3
# #
# |
| 36 | 17, 35 | eqtr3d 2224 |
. 2
# #
#
|
| 37 | 6, 1 | mulcld 7997 |
. . . 4
# #
# |
| 38 | 7, 2 | mulcld 7997 |
. . . 4
# #
# |
| 39 | mulap0 8630 |
. . . . 5
#
# #
| |
| 40 | 39 | ad2ant2lr 510 |
. . . 4
# #
# #
|
| 41 | divclap 8654 |
. . . 4
#
| |
| 42 | 37, 38, 40, 41 | syl3anc 1249 |
. . 3
# #
# |
| 43 | divap0 8660 |
. . . 4
#
# #
| |
| 44 | 43 | adantl 277 |
. . 3
# #
# #
|
| 45 | divmulap 8651 |
. . 3
#
| |
| 46 | 10, 42, 32, 44, 45 | syl112anc 1253 |
. 2
# #
#
|
| 47 | 36, 46 | mpbird 167 |
1
# #
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wceq 1364 wcel 2160 class class class wbr 4018
(class class class)co 5891 cc 7828 cc0 7830 c1 7831 cmul 7835 # cap 8557
cdiv 8648 |
| 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 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-mulrcl 7929 ax-addcom 7930 ax-mulcom 7931 ax-addass 7932 ax-mulass 7933 ax-distr 7934 ax-i2m1 7935 ax-0lt1 7936 ax-1rid 7937 ax-0id 7938 ax-rnegex 7939 ax-precex 7940 ax-cnre 7941 ax-pre-ltirr 7942 ax-pre-ltwlin 7943 ax-pre-lttrn 7944 ax-pre-apti 7945 ax-pre-ltadd 7946 ax-pre-mulgt0 7947 ax-pre-mulext 7948 |
| 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 4308 df-po 4311 df-iso 4312 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-iota 5193 df-fun 5233 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-le 8017 df-sub 8149 df-neg 8150 df-reap 8551 df-ap 8558 df-div 8649 |
| This theorem is referenced by: recdivap 8694 divcanap7 8697 divdivap1 8699 divdivap2 8700 divdivdivapi 8751 qreccl 9661 pcadd 12358 |
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