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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 534 |
. . . . . . 7
# #
# | |
| 2 | simprll 532 |
. . . . . . 7
# #
# | |
| 3 | simprlr 533 |
. . . . . . 7
# #
# # | |
| 4 | divclap 8595 |
. . . . . . 7
# | |
| 5 | 1, 2, 3, 4 | syl3anc 1233 |
. . . . . 6
# #
# |
| 6 | simpll 524 |
. . . . . . 7
# #
# | |
| 7 | simplrl 530 |
. . . . . . 7
# #
# | |
| 8 | simplrr 531 |
. . . . . . 7
# #
# # | |
| 9 | divclap 8595 |
. . . . . . 7
# | |
| 10 | 6, 7, 8, 9 | syl3anc 1233 |
. . . . . 6
# #
# |
| 11 | 5, 10 | mulcomd 7941 |
. . . . 5
# #
# |
| 12 | simplr 525 |
. . . . . 6
# #
# # | |
| 13 | simprl 526 |
. . . . . 6
# #
# # | |
| 14 | divmuldivap 8629 |
. . . . . 6
#
# | |
| 15 | 6, 1, 12, 13, 14 | syl22anc 1234 |
. . . . 5
# #
# |
| 16 | 11, 15 | eqtrd 2203 |
. . . 4
# #
# |
| 17 | 16 | oveq2d 5869 |
. . 3
# #
# |
| 18 | simprr 527 |
. . . . . . 7
# #
# # | |
| 19 | divmuldivap 8629 |
. . . . . . 7
#
# | |
| 20 | 2, 1, 18, 13, 19 | syl22anc 1234 |
. . . . . 6
# #
# |
| 21 | 2, 1 | mulcomd 7941 |
. . . . . . . 8
# #
# |
| 22 | 21 | oveq1d 5868 |
. . . . . . 7
# #
# |
| 23 | 1, 2 | mulcld 7940 |
. . . . . . . 8
# #
# |
| 24 | simprrr 535 |
. . . . . . . . 9
# #
# # | |
| 25 | 1, 2, 24, 3 | mulap0d 8576 |
. . . . . . . 8
# #
# #
|
| 26 | dividap 8618 |
. . . . . . . 8
#
 | |
| 27 | 23, 25, 26 | syl2anc 409 |
. . . . . . 7
# #
# |
| 28 | 22, 27 | eqtrd 2203 |
. . . . . 6
# #
# |
| 29 | 20, 28 | eqtrd 2203 |
. . . . 5
# #
# |
| 30 | 29 | oveq1d 5868 |
. . . 4
# #
# |
| 31 | divclap 8595 |
. . . . . 6
# | |
| 32 | 2, 1, 24, 31 | syl3anc 1233 |
. . . . 5
# #
# |
| 33 | 32, 5, 10 | mulassd 7943 |
. . . 4
# #
# |
| 34 | 10 | mulid2d 7938 |
. . . 4
# #
# |
| 35 | 30, 33, 34 | 3eqtr3d 2211 |
. . 3
# #
# |
| 36 | 17, 35 | eqtr3d 2205 |
. 2
# #
#
|
| 37 | 6, 1 | mulcld 7940 |
. . . 4
# #
# |
| 38 | 7, 2 | mulcld 7940 |
. . . 4
# #
# |
| 39 | mulap0 8572 |
. . . . 5
#
# #
| |
| 40 | 39 | ad2ant2lr 507 |
. . . 4
# #
# #
|
| 41 | divclap 8595 |
. . . 4
#
| |
| 42 | 37, 38, 40, 41 | syl3anc 1233 |
. . 3
# #
# |
| 43 | divap0 8601 |
. . . 4
#
# #
| |
| 44 | 43 | adantl 275 |
. . 3
# #
# #
|
| 45 | divmulap 8592 |
. . 3
#
| |
| 46 | 10, 42, 32, 44, 45 | syl112anc 1237 |
. 2
# #
#
|
| 47 | 36, 46 | mpbird 166 |
1
# #
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 wceq 1348 wcel 2141 class class class wbr 3989
(class class class)co 5853 cc 7772 cc0 7774 c1 7775 cmul 7779 # cap 8500
cdiv 8589 |
| 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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 |
| This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-po 4281 df-iso 4282 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 |
| This theorem is referenced by: recdivap 8635 divcanap7 8638 divdivap1 8640 divdivap2 8641 divdivdivapi 8692 qreccl 9601 pcadd 12293 |
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