Nearby theorems
|
|
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
| 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 507 |
. . . . . . 7
# #
# | |
| 2 | simprll 505 |
. . . . . . 7
# #
# | |
| 3 | simprlr 506 |
. . . . . . 7
# #
# # | |
| 4 | divclap 8199 |
. . . . . . 7
# | |
| 5 | 1, 2, 3, 4 | syl3anc 1175 |
. . . . . 6
# #
# |
| 6 | simpll 497 |
. . . . . . 7
# #
# | |
| 7 | simplrl 503 |
. . . . . . 7
# #
# | |
| 8 | simplrr 504 |
. . . . . . 7
# #
# # | |
| 9 | divclap 8199 |
. . . . . . 7
# | |
| 10 | 6, 7, 8, 9 | syl3anc 1175 |
. . . . . 6
# #
# |
| 11 | 5, 10 | mulcomd 7563 |
. . . . 5
# #
# |
| 12 | simplr 498 |
. . . . . 6
# #
# # | |
| 13 | simprl 499 |
. . . . . 6
# #
# # | |
| 14 | divmuldivap 8233 |
. . . . . 6
#
# | |
| 15 | 6, 1, 12, 13, 14 | syl22anc 1176 |
. . . . 5
# #
# |
| 16 | 11, 15 | eqtrd 2121 |
. . . 4
# #
# |
| 17 | 16 | oveq2d 5682 |
. . 3
# #
# |
| 18 | simprr 500 |
. . . . . . 7
# #
# # | |
| 19 | divmuldivap 8233 |
. . . . . . 7
#
# | |
| 20 | 2, 1, 18, 13, 19 | syl22anc 1176 |
. . . . . 6
# #
# |
| 21 | 2, 1 | mulcomd 7563 |
. . . . . . . 8
# #
# |
| 22 | 21 | oveq1d 5681 |
. . . . . . 7
# #
# |
| 23 | 1, 2 | mulcld 7562 |
. . . . . . . 8
# #
# |
| 24 | simprrr 508 |
. . . . . . . . 9
# #
# # | |
| 25 | 1, 2, 24, 3 | mulap0d 8181 |
. . . . . . . 8
# #
# #
|
| 26 | dividap 8222 |
. . . . . . . 8
#
 | |
| 27 | 23, 25, 26 | syl2anc 404 |
. . . . . . 7
# #
# |
| 28 | 22, 27 | eqtrd 2121 |
. . . . . 6
# #
# |
| 29 | 20, 28 | eqtrd 2121 |
. . . . 5
# #
# |
| 30 | 29 | oveq1d 5681 |
. . . 4
# #
# |
| 31 | divclap 8199 |
. . . . . 6
# | |
| 32 | 2, 1, 24, 31 | syl3anc 1175 |
. . . . 5
# #
# |
| 33 | 32, 5, 10 | mulassd 7565 |
. . . 4
# #
# |
| 34 | 10 | mulid2d 7560 |
. . . 4
# #
# |
| 35 | 30, 33, 34 | 3eqtr3d 2129 |
. . 3
# #
# |
| 36 | 17, 35 | eqtr3d 2123 |
. 2
# #
#
|
| 37 | 6, 1 | mulcld 7562 |
. . . 4
# #
# |
| 38 | 7, 2 | mulcld 7562 |
. . . 4
# #
# |
| 39 | mulap0 8177 |
. . . . 5
#
# #
| |
| 40 | 39 | ad2ant2lr 495 |
. . . 4
# #
# #
|
| 41 | divclap 8199 |
. . . 4
#
| |
| 42 | 37, 38, 40, 41 | syl3anc 1175 |
. . 3
# #
# |
| 43 | divap0 8205 |
. . . 4
#
# #
| |
| 44 | 43 | adantl 272 |
. . 3
# #
# #
|
| 45 | divmulap 8196 |
. . 3
#
| |
| 46 | 10, 42, 32, 44, 45 | syl112anc 1179 |
. 2
# #
#
|
| 47 | 36, 46 | mpbird 166 |
1
# #
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 wceq 1290 wcel 1439 class class class wbr 3851
(class class class)co 5666 cc 7402 cc0 7404 c1 7405 cmul 7409 # cap 8112
cdiv 8193 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7490 ax-resscn 7491 ax-1cn 7492 ax-1re 7493 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-mulrcl 7498 ax-addcom 7499 ax-mulcom 7500 ax-addass 7501 ax-mulass 7502 ax-distr 7503 ax-i2m1 7504 ax-0lt1 7505 ax-1rid 7506 ax-0id 7507 ax-rnegex 7508 ax-precex 7509 ax-cnre 7510 ax-pre-ltirr 7511 ax-pre-ltwlin 7512 ax-pre-lttrn 7513 ax-pre-apti 7514 ax-pre-ltadd 7515 ax-pre-mulgt0 7516 ax-pre-mulext 7517 |
| This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-id 4129 df-po 4132 df-iso 4133 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-pnf 7578 df-mnf 7579 df-xr 7580 df-ltxr 7581 df-le 7582 df-sub 7709 df-neg 7710 df-reap 8106 df-ap 8113 df-div 8194 |
| This theorem is referenced by: recdivap 8239 divcanap7 8242 divdivap1 8244 divdivap2 8245 divdivdivapi 8296 qreccl 9181 |
| Copyright terms: Public domain | W3C validator |