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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 8705 |
. . . . . . 7
# | |
| 5 | 1, 2, 3, 4 | syl3anc 1249 |
. . . . . 6
# #
# |
| 6 | simpll 527 |
. . . . . . 7
# #
# | |
| 7 | simplrl 535 |
. . . . . . 7
# #
# | |
| 8 | simplrr 536 |
. . . . . . 7
# #
# # | |
| 9 | divclap 8705 |
. . . . . . 7
# | |
| 10 | 6, 7, 8, 9 | syl3anc 1249 |
. . . . . 6
# #
# |
| 11 | 5, 10 | mulcomd 8048 |
. . . . 5
# #
# |
| 12 | simplr 528 |
. . . . . 6
# #
# # | |
| 13 | simprl 529 |
. . . . . 6
# #
# # | |
| 14 | divmuldivap 8739 |
. . . . . 6
#
# | |
| 15 | 6, 1, 12, 13, 14 | syl22anc 1250 |
. . . . 5
# #
# |
| 16 | 11, 15 | eqtrd 2229 |
. . . 4
# #
# |
| 17 | 16 | oveq2d 5938 |
. . 3
# #
# |
| 18 | simprr 531 |
. . . . . . 7
# #
# # | |
| 19 | divmuldivap 8739 |
. . . . . . 7
#
# | |
| 20 | 2, 1, 18, 13, 19 | syl22anc 1250 |
. . . . . 6
# #
# |
| 21 | 2, 1 | mulcomd 8048 |
. . . . . . . 8
# #
# |
| 22 | 21 | oveq1d 5937 |
. . . . . . 7
# #
# |
| 23 | 1, 2 | mulcld 8047 |
. . . . . . . 8
# #
# |
| 24 | simprrr 540 |
. . . . . . . . 9
# #
# # | |
| 25 | 1, 2, 24, 3 | mulap0d 8685 |
. . . . . . . 8
# #
# #
|
| 26 | dividap 8728 |
. . . . . . . 8
#
 | |
| 27 | 23, 25, 26 | syl2anc 411 |
. . . . . . 7
# #
# |
| 28 | 22, 27 | eqtrd 2229 |
. . . . . 6
# #
# |
| 29 | 20, 28 | eqtrd 2229 |
. . . . 5
# #
# |
| 30 | 29 | oveq1d 5937 |
. . . 4
# #
# |
| 31 | divclap 8705 |
. . . . . 6
# | |
| 32 | 2, 1, 24, 31 | syl3anc 1249 |
. . . . 5
# #
# |
| 33 | 32, 5, 10 | mulassd 8050 |
. . . 4
# #
# |
| 34 | 10 | mulid2d 8045 |
. . . 4
# #
# |
| 35 | 30, 33, 34 | 3eqtr3d 2237 |
. . 3
# #
# |
| 36 | 17, 35 | eqtr3d 2231 |
. 2
# #
#
|
| 37 | 6, 1 | mulcld 8047 |
. . . 4
# #
# |
| 38 | 7, 2 | mulcld 8047 |
. . . 4
# #
# |
| 39 | mulap0 8681 |
. . . . 5
#
# #
| |
| 40 | 39 | ad2ant2lr 510 |
. . . 4
# #
# #
|
| 41 | divclap 8705 |
. . . 4
#
| |
| 42 | 37, 38, 40, 41 | syl3anc 1249 |
. . 3
# #
# |
| 43 | divap0 8711 |
. . . 4
#
# #
| |
| 44 | 43 | adantl 277 |
. . 3
# #
# #
|
| 45 | divmulap 8702 |
. . 3
#
| |
| 46 | 10, 42, 32, 44, 45 | syl112anc 1253 |
. 2
# #
#
|
| 47 | 36, 46 | mpbird 167 |
1
# #
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wceq 1364 wcel 2167 class class class wbr 4033
(class class class)co 5922 cc 7877 cc0 7879 c1 7880 cmul 7884 # cap 8608
cdiv 8699 |
| 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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-br 4034 df-opab 4095 df-id 4328 df-po 4331 df-iso 4332 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-iota 5219 df-fun 5260 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 |
| This theorem is referenced by: recdivap 8745 divcanap7 8748 divdivap1 8750 divdivap2 8751 divdivdivapi 8802 qreccl 9716 pcadd 12509 |
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