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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 8462 |
. . . . . . 7
# | |
| 5 | 1, 2, 3, 4 | syl3anc 1217 |
. . . . . 6
# #
# |
| 6 | simpll 519 |
. . . . . . 7
# #
# | |
| 7 | simplrl 525 |
. . . . . . 7
# #
# | |
| 8 | simplrr 526 |
. . . . . . 7
# #
# # | |
| 9 | divclap 8462 |
. . . . . . 7
# | |
| 10 | 6, 7, 8, 9 | syl3anc 1217 |
. . . . . 6
# #
# |
| 11 | 5, 10 | mulcomd 7811 |
. . . . 5
# #
# |
| 12 | simplr 520 |
. . . . . 6
# #
# # | |
| 13 | simprl 521 |
. . . . . 6
# #
# # | |
| 14 | divmuldivap 8496 |
. . . . . 6
#
# | |
| 15 | 6, 1, 12, 13, 14 | syl22anc 1218 |
. . . . 5
# #
# |
| 16 | 11, 15 | eqtrd 2173 |
. . . 4
# #
# |
| 17 | 16 | oveq2d 5798 |
. . 3
# #
# |
| 18 | simprr 522 |
. . . . . . 7
# #
# # | |
| 19 | divmuldivap 8496 |
. . . . . . 7
#
# | |
| 20 | 2, 1, 18, 13, 19 | syl22anc 1218 |
. . . . . 6
# #
# |
| 21 | 2, 1 | mulcomd 7811 |
. . . . . . . 8
# #
# |
| 22 | 21 | oveq1d 5797 |
. . . . . . 7
# #
# |
| 23 | 1, 2 | mulcld 7810 |
. . . . . . . 8
# #
# |
| 24 | simprrr 530 |
. . . . . . . . 9
# #
# # | |
| 25 | 1, 2, 24, 3 | mulap0d 8443 |
. . . . . . . 8
# #
# #
|
| 26 | dividap 8485 |
. . . . . . . 8
#
 | |
| 27 | 23, 25, 26 | syl2anc 409 |
. . . . . . 7
# #
# |
| 28 | 22, 27 | eqtrd 2173 |
. . . . . 6
# #
# |
| 29 | 20, 28 | eqtrd 2173 |
. . . . 5
# #
# |
| 30 | 29 | oveq1d 5797 |
. . . 4
# #
# |
| 31 | divclap 8462 |
. . . . . 6
# | |
| 32 | 2, 1, 24, 31 | syl3anc 1217 |
. . . . 5
# #
# |
| 33 | 32, 5, 10 | mulassd 7813 |
. . . 4
# #
# |
| 34 | 10 | mulid2d 7808 |
. . . 4
# #
# |
| 35 | 30, 33, 34 | 3eqtr3d 2181 |
. . 3
# #
# |
| 36 | 17, 35 | eqtr3d 2175 |
. 2
# #
#
|
| 37 | 6, 1 | mulcld 7810 |
. . . 4
# #
# |
| 38 | 7, 2 | mulcld 7810 |
. . . 4
# #
# |
| 39 | mulap0 8439 |
. . . . 5
#
# #
| |
| 40 | 39 | ad2ant2lr 502 |
. . . 4
# #
# #
|
| 41 | divclap 8462 |
. . . 4
#
| |
| 42 | 37, 38, 40, 41 | syl3anc 1217 |
. . 3
# #
# |
| 43 | divap0 8468 |
. . . 4
#
# #
| |
| 44 | 43 | adantl 275 |
. . 3
# #
# #
|
| 45 | divmulap 8459 |
. . 3
#
| |
| 46 | 10, 42, 32, 44, 45 | syl112anc 1221 |
. 2
# #
#
|
| 47 | 36, 46 | mpbird 166 |
1
# #
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 wceq 1332 wcel 1481 class class class wbr 3937
(class class class)co 5782 cc 7642 cc0 7644 c1 7645 cmul 7649 # cap 8367
cdiv 8456 |
| 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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 |
| This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-po 4226 df-iso 4227 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 |
| This theorem is referenced by: recdivap 8502 divcanap7 8505 divdivap1 8507 divdivap2 8508 divdivdivapi 8559 qreccl 9461 |
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