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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 529 |
. . . . . . 7
# #
# | |
| 2 | simprll 527 |
. . . . . . 7
# #
# | |
| 3 | simprlr 528 |
. . . . . . 7
# #
# # | |
| 4 | divclap 8574 |
. . . . . . 7
# | |
| 5 | 1, 2, 3, 4 | syl3anc 1228 |
. . . . . 6
# #
# |
| 6 | simpll 519 |
. . . . . . 7
# #
# | |
| 7 | simplrl 525 |
. . . . . . 7
# #
# | |
| 8 | simplrr 526 |
. . . . . . 7
# #
# # | |
| 9 | divclap 8574 |
. . . . . . 7
# | |
| 10 | 6, 7, 8, 9 | syl3anc 1228 |
. . . . . 6
# #
# |
| 11 | 5, 10 | mulcomd 7920 |
. . . . 5
# #
# |
| 12 | simplr 520 |
. . . . . 6
# #
# # | |
| 13 | simprl 521 |
. . . . . 6
# #
# # | |
| 14 | divmuldivap 8608 |
. . . . . 6
#
# | |
| 15 | 6, 1, 12, 13, 14 | syl22anc 1229 |
. . . . 5
# #
# |
| 16 | 11, 15 | eqtrd 2198 |
. . . 4
# #
# |
| 17 | 16 | oveq2d 5858 |
. . 3
# #
# |
| 18 | simprr 522 |
. . . . . . 7
# #
# # | |
| 19 | divmuldivap 8608 |
. . . . . . 7
#
# | |
| 20 | 2, 1, 18, 13, 19 | syl22anc 1229 |
. . . . . 6
# #
# |
| 21 | 2, 1 | mulcomd 7920 |
. . . . . . . 8
# #
# |
| 22 | 21 | oveq1d 5857 |
. . . . . . 7
# #
# |
| 23 | 1, 2 | mulcld 7919 |
. . . . . . . 8
# #
# |
| 24 | simprrr 530 |
. . . . . . . . 9
# #
# # | |
| 25 | 1, 2, 24, 3 | mulap0d 8555 |
. . . . . . . 8
# #
# #
|
| 26 | dividap 8597 |
. . . . . . . 8
#
 | |
| 27 | 23, 25, 26 | syl2anc 409 |
. . . . . . 7
# #
# |
| 28 | 22, 27 | eqtrd 2198 |
. . . . . 6
# #
# |
| 29 | 20, 28 | eqtrd 2198 |
. . . . 5
# #
# |
| 30 | 29 | oveq1d 5857 |
. . . 4
# #
# |
| 31 | divclap 8574 |
. . . . . 6
# | |
| 32 | 2, 1, 24, 31 | syl3anc 1228 |
. . . . 5
# #
# |
| 33 | 32, 5, 10 | mulassd 7922 |
. . . 4
# #
# |
| 34 | 10 | mulid2d 7917 |
. . . 4
# #
# |
| 35 | 30, 33, 34 | 3eqtr3d 2206 |
. . 3
# #
# |
| 36 | 17, 35 | eqtr3d 2200 |
. 2
# #
#
|
| 37 | 6, 1 | mulcld 7919 |
. . . 4
# #
# |
| 38 | 7, 2 | mulcld 7919 |
. . . 4
# #
# |
| 39 | mulap0 8551 |
. . . . 5
#
# #
| |
| 40 | 39 | ad2ant2lr 502 |
. . . 4
# #
# #
|
| 41 | divclap 8574 |
. . . 4
#
| |
| 42 | 37, 38, 40, 41 | syl3anc 1228 |
. . 3
# #
# |
| 43 | divap0 8580 |
. . . 4
#
# #
| |
| 44 | 43 | adantl 275 |
. . 3
# #
# #
|
| 45 | divmulap 8571 |
. . 3
#
| |
| 46 | 10, 42, 32, 44, 45 | syl112anc 1232 |
. 2
# #
#
|
| 47 | 36, 46 | mpbird 166 |
1
# #
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 wceq 1343 wcel 2136 class class class wbr 3982
(class class class)co 5842 cc 7751 cc0 7753 c1 7754 cmul 7758 # cap 8479
cdiv 8568 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 |
| This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-id 4271 df-po 4274 df-iso 4275 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-iota 5153 df-fun 5190 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 |
| This theorem is referenced by: recdivap 8614 divcanap7 8617 divdivap1 8619 divdivap2 8620 divdivdivapi 8671 qreccl 9580 pcadd 12271 |
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