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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 985 |
. . 3
# 
# | |
| 2 | 3anass 985 |
. . 3
#
# | |
| 3 | divclap 8786 |
. . . . . 6
# | |
| 4 | divclap 8786 |
. . . . . 6
# | |
| 5 | mulcl 8087 |
. . . . . 6
| |
| 6 | 3, 4, 5 | syl2an 289 |
. . . . 5
#
# |
| 7 | mulcl 8087 |
. . . . . . . 8
| |
| 8 | 7 | ad2ant2r 509 |
. . . . . . 7
#
# |
| 9 | 8 | 3adantr1 1159 |
. . . . . 6
#
# |
| 10 | 9 | 3adantl1 1156 |
. . . . 5
#
#
|
| 11 | mulap0 8762 |
. . . . . . 7
#
# #
| |
| 12 | 11 | 3adantr1 1159 |
. . . . . 6
#
# # |
| 13 | 12 | 3adantl1 1156 |
. . . . 5
#
#
# |
| 14 | divcanap3 8806 |
. . . . 5
#
| |
| 15 | 6, 10, 13, 14 | syl3anc 1250 |
. . . 4
#
# |
| 16 | simp2 1001 |
. . . . . . . 8
# | |
| 17 | 16, 3 | jca 306 |
. . . . . . 7
#
|
| 18 | simp2 1001 |
. . . . . . . 8
# | |
| 19 | 18, 4 | jca 306 |
. . . . . . 7
#
|
| 20 | mul4 8239 |
. . . . . . 7
| |
| 21 | 17, 19, 20 | syl2an 289 |
. . . . . 6
#
#
|
| 22 | divcanap2 8788 |
. . . . . . 7
#
| |
| 23 | divcanap2 8788 |
. . . . . . 7
#
| |
| 24 | 22, 23 | oveqan12d 5986 |
. . . . . 6
#
#
|
| 25 | 21, 24 | eqtr3d 2242 |
. . . . 5
#
# |
| 26 | 25 | oveq1d 5982 |
. . . 4
#
# |
| 27 | 15, 26 | eqtr3d 2242 |
. . 3
#
# |
| 28 | 1, 2, 27 | syl2anbr 292 |
. 2
# # |
| 29 | 28 | an4s 588 |
1
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
w3a 981
wceq 1373
wcel 2178
class class class wbr 4059 (class class class)co 5967
cc 7958
cc0 7960
cmul 7965
# cap 8689 cdiv 8780 |
| 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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-sep 4178 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-cnex 8051 ax-resscn 8052 ax-1cn 8053 ax-1re 8054 ax-icn 8055 ax-addcl 8056 ax-addrcl 8057 ax-mulcl 8058 ax-mulrcl 8059 ax-addcom 8060 ax-mulcom 8061 ax-addass 8062 ax-mulass 8063 ax-distr 8064 ax-i2m1 8065 ax-0lt1 8066 ax-1rid 8067 ax-0id 8068 ax-rnegex 8069 ax-precex 8070 ax-cnre 8071 ax-pre-ltirr 8072 ax-pre-ltwlin 8073 ax-pre-lttrn 8074 ax-pre-apti 8075 ax-pre-ltadd 8076 ax-pre-mulgt0 8077 ax-pre-mulext 8078 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-reu 2493 df-rmo 2494 df-rab 2495 df-v 2778 df-sbc 3006 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-br 4060 df-opab 4122 df-id 4358 df-po 4361 df-iso 4362 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-iota 5251 df-fun 5292 df-fv 5298 df-riota 5922 df-ov 5970 df-oprab 5971 df-mpo 5972 df-pnf 8144 df-mnf 8145 df-xr 8146 df-ltxr 8147 df-le 8148 df-sub 8280 df-neg 8281 df-reap 8683 df-ap 8690 df-div 8781 |
| This theorem is referenced by: divdivdivap 8821 divcanap5 8822 divmul13ap 8823 divmul24ap 8824 divmuldivapi 8880 divmuldivapd 8940 qmulcl 9793 mulexpzap 10761 expaddzap 10765 sqdivap 10785 |
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