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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 528 |
. . . . . . 7
# #
# | |
| 2 | simprll 526 |
. . . . . . 7
# #
# | |
| 3 | simprlr 527 |
. . . . . . 7
# #
# # | |
| 4 | divclap 8431 |
. . . . . . 7
# | |
| 5 | 1, 2, 3, 4 | syl3anc 1216 |
. . . . . 6
# #
# |
| 6 | simpll 518 |
. . . . . . 7
# #
# | |
| 7 | simplrl 524 |
. . . . . . 7
# #
# | |
| 8 | simplrr 525 |
. . . . . . 7
# #
# # | |
| 9 | divclap 8431 |
. . . . . . 7
# | |
| 10 | 6, 7, 8, 9 | syl3anc 1216 |
. . . . . 6
# #
# |
| 11 | 5, 10 | mulcomd 7780 |
. . . . 5
# #
# |
| 12 | simplr 519 |
. . . . . 6
# #
# # | |
| 13 | simprl 520 |
. . . . . 6
# #
# # | |
| 14 | divmuldivap 8465 |
. . . . . 6
#
# | |
| 15 | 6, 1, 12, 13, 14 | syl22anc 1217 |
. . . . 5
# #
# |
| 16 | 11, 15 | eqtrd 2170 |
. . . 4
# #
# |
| 17 | 16 | oveq2d 5783 |
. . 3
# #
# |
| 18 | simprr 521 |
. . . . . . 7
# #
# # | |
| 19 | divmuldivap 8465 |
. . . . . . 7
#
# | |
| 20 | 2, 1, 18, 13, 19 | syl22anc 1217 |
. . . . . 6
# #
# |
| 21 | 2, 1 | mulcomd 7780 |
. . . . . . . 8
# #
# |
| 22 | 21 | oveq1d 5782 |
. . . . . . 7
# #
# |
| 23 | 1, 2 | mulcld 7779 |
. . . . . . . 8
# #
# |
| 24 | simprrr 529 |
. . . . . . . . 9
# #
# # | |
| 25 | 1, 2, 24, 3 | mulap0d 8412 |
. . . . . . . 8
# #
# #
|
| 26 | dividap 8454 |
. . . . . . . 8
#
 | |
| 27 | 23, 25, 26 | syl2anc 408 |
. . . . . . 7
# #
# |
| 28 | 22, 27 | eqtrd 2170 |
. . . . . 6
# #
# |
| 29 | 20, 28 | eqtrd 2170 |
. . . . 5
# #
# |
| 30 | 29 | oveq1d 5782 |
. . . 4
# #
# |
| 31 | divclap 8431 |
. . . . . 6
# | |
| 32 | 2, 1, 24, 31 | syl3anc 1216 |
. . . . 5
# #
# |
| 33 | 32, 5, 10 | mulassd 7782 |
. . . 4
# #
# |
| 34 | 10 | mulid2d 7777 |
. . . 4
# #
# |
| 35 | 30, 33, 34 | 3eqtr3d 2178 |
. . 3
# #
# |
| 36 | 17, 35 | eqtr3d 2172 |
. 2
# #
#
|
| 37 | 6, 1 | mulcld 7779 |
. . . 4
# #
# |
| 38 | 7, 2 | mulcld 7779 |
. . . 4
# #
# |
| 39 | mulap0 8408 |
. . . . 5
#
# #
| |
| 40 | 39 | ad2ant2lr 501 |
. . . 4
# #
# #
|
| 41 | divclap 8431 |
. . . 4
#
| |
| 42 | 37, 38, 40, 41 | syl3anc 1216 |
. . 3
# #
# |
| 43 | divap0 8437 |
. . . 4
#
# #
| |
| 44 | 43 | adantl 275 |
. . 3
# #
# #
|
| 45 | divmulap 8428 |
. . 3
#
| |
| 46 | 10, 42, 32, 44, 45 | syl112anc 1220 |
. 2
# #
#
|
| 47 | 36, 46 | mpbird 166 |
1
# #
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 wceq 1331 wcel 1480 class class class wbr 3924
(class class class)co 5767 cc 7611 cc0 7613 c1 7614 cmul 7618 # cap 8336
cdiv 8425 |
| 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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 |
| This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 |
| This theorem is referenced by: recdivap 8471 divcanap7 8474 divdivap1 8476 divdivap2 8477 divdivdivapi 8528 qreccl 9427 |
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