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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 8648 |
. . . . . . 7
# | |
| 5 | 1, 2, 3, 4 | syl3anc 1248 |
. . . . . 6
# #
# |
| 6 | simpll 527 |
. . . . . . 7
# #
# | |
| 7 | simplrl 535 |
. . . . . . 7
# #
# | |
| 8 | simplrr 536 |
. . . . . . 7
# #
# # | |
| 9 | divclap 8648 |
. . . . . . 7
# | |
| 10 | 6, 7, 8, 9 | syl3anc 1248 |
. . . . . 6
# #
# |
| 11 | 5, 10 | mulcomd 7992 |
. . . . 5
# #
# |
| 12 | simplr 528 |
. . . . . 6
# #
# # | |
| 13 | simprl 529 |
. . . . . 6
# #
# # | |
| 14 | divmuldivap 8682 |
. . . . . 6
#
# | |
| 15 | 6, 1, 12, 13, 14 | syl22anc 1249 |
. . . . 5
# #
# |
| 16 | 11, 15 | eqtrd 2220 |
. . . 4
# #
# |
| 17 | 16 | oveq2d 5904 |
. . 3
# #
# |
| 18 | simprr 531 |
. . . . . . 7
# #
# # | |
| 19 | divmuldivap 8682 |
. . . . . . 7
#
# | |
| 20 | 2, 1, 18, 13, 19 | syl22anc 1249 |
. . . . . 6
# #
# |
| 21 | 2, 1 | mulcomd 7992 |
. . . . . . . 8
# #
# |
| 22 | 21 | oveq1d 5903 |
. . . . . . 7
# #
# |
| 23 | 1, 2 | mulcld 7991 |
. . . . . . . 8
# #
# |
| 24 | simprrr 540 |
. . . . . . . . 9
# #
# # | |
| 25 | 1, 2, 24, 3 | mulap0d 8628 |
. . . . . . . 8
# #
# #
|
| 26 | dividap 8671 |
. . . . . . . 8
#
 | |
| 27 | 23, 25, 26 | syl2anc 411 |
. . . . . . 7
# #
# |
| 28 | 22, 27 | eqtrd 2220 |
. . . . . 6
# #
# |
| 29 | 20, 28 | eqtrd 2220 |
. . . . 5
# #
# |
| 30 | 29 | oveq1d 5903 |
. . . 4
# #
# |
| 31 | divclap 8648 |
. . . . . 6
# | |
| 32 | 2, 1, 24, 31 | syl3anc 1248 |
. . . . 5
# #
# |
| 33 | 32, 5, 10 | mulassd 7994 |
. . . 4
# #
# |
| 34 | 10 | mulid2d 7989 |
. . . 4
# #
# |
| 35 | 30, 33, 34 | 3eqtr3d 2228 |
. . 3
# #
# |
| 36 | 17, 35 | eqtr3d 2222 |
. 2
# #
#
|
| 37 | 6, 1 | mulcld 7991 |
. . . 4
# #
# |
| 38 | 7, 2 | mulcld 7991 |
. . . 4
# #
# |
| 39 | mulap0 8624 |
. . . . 5
#
# #
| |
| 40 | 39 | ad2ant2lr 510 |
. . . 4
# #
# #
|
| 41 | divclap 8648 |
. . . 4
#
| |
| 42 | 37, 38, 40, 41 | syl3anc 1248 |
. . 3
# #
# |
| 43 | divap0 8654 |
. . . 4
#
# #
| |
| 44 | 43 | adantl 277 |
. . 3
# #
# #
|
| 45 | divmulap 8645 |
. . 3
#
| |
| 46 | 10, 42, 32, 44, 45 | syl112anc 1252 |
. 2
# #
#
|
| 47 | 36, 46 | mpbird 167 |
1
# #
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wceq 1363 wcel 2158 class class class wbr 4015
(class class class)co 5888 cc 7822 cc0 7824 c1 7825 cmul 7829 # cap 8551
cdiv 8642 |
| 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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942 |
| This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-id 4305 df-po 4308 df-iso 4309 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 |
| This theorem is referenced by: recdivap 8688 divcanap7 8691 divdivap1 8693 divdivap2 8694 divdivdivapi 8745 qreccl 9655 pcadd 12352 |
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