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| Mirrors > Home > ILE Home > Th. List > divmuldivap | Unicode version | ||
| Description: Multiplication of two ratios. (Contributed by Jim Kingdon, 25-Feb-2020.) |
| Ref | Expression |
|---|---|
| divmuldivap |
     ![/\](wedge.gif "/\"){kind=small}    #  
#     |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3anass 967 |
. . 3
# 
# | |
| 2 | 3anass 967 |
. . 3
#
# | |
| 3 | divclap 8462 |
. . . . . 6
# | |
| 4 | divclap 8462 |
. . . . . 6
# | |
| 5 | mulcl 7771 |
. . . . . 6
| |
| 6 | 3, 4, 5 | syl2an 287 |
. . . . 5
#
# |
| 7 | mulcl 7771 |
. . . . . . . 8
| |
| 8 | 7 | ad2ant2r 501 |
. . . . . . 7
#
# |
| 9 | 8 | 3adantr1 1141 |
. . . . . 6
#
# |
| 10 | 9 | 3adantl1 1138 |
. . . . 5
#
#
|
| 11 | mulap0 8439 |
. . . . . . 7
#
# #
| |
| 12 | 11 | 3adantr1 1141 |
. . . . . 6
#
# # |
| 13 | 12 | 3adantl1 1138 |
. . . . 5
#
#
# |
| 14 | divcanap3 8482 |
. . . . 5
#
| |
| 15 | 6, 10, 13, 14 | syl3anc 1217 |
. . . 4
#
# |
| 16 | simp2 983 |
. . . . . . . 8
# | |
| 17 | 16, 3 | jca 304 |
. . . . . . 7
#
|
| 18 | simp2 983 |
. . . . . . . 8
# | |
| 19 | 18, 4 | jca 304 |
. . . . . . 7
#
|
| 20 | mul4 7918 |
. . . . . . 7
| |
| 21 | 17, 19, 20 | syl2an 287 |
. . . . . 6
#
#
|
| 22 | divcanap2 8464 |
. . . . . . 7
#
| |
| 23 | divcanap2 8464 |
. . . . . . 7
#
| |
| 24 | 22, 23 | oveqan12d 5801 |
. . . . . 6
#
#
|
| 25 | 21, 24 | eqtr3d 2175 |
. . . . 5
#
# |
| 26 | 25 | oveq1d 5797 |
. . . 4
#
# |
| 27 | 15, 26 | eqtr3d 2175 |
. . 3
#
# |
| 28 | 1, 2, 27 | syl2anbr 290 |
. 2
# # |
| 29 | 28 | an4s 578 |
1
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
w3a 963
wceq 1332
wcel 1481
class class class wbr 3937 (class class class)co 5782
cc 7642
cc0 7644
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: divdivdivap 8497 divcanap5 8498 divmul13ap 8499 divmul24ap 8500 divmuldivapi 8556 divmuldivapd 8616 qmulcl 9456 mulexpzap 10364 expaddzap 10368 sqdivap 10388 |
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