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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 984 |
. . 3
# 
# | |
| 2 | 3anass 984 |
. . 3
#
# | |
| 3 | divclap 8699 |
. . . . . 6
# | |
| 4 | divclap 8699 |
. . . . . 6
# | |
| 5 | mulcl 8001 |
. . . . . 6
| |
| 6 | 3, 4, 5 | syl2an 289 |
. . . . 5
#
# |
| 7 | mulcl 8001 |
. . . . . . . 8
| |
| 8 | 7 | ad2ant2r 509 |
. . . . . . 7
#
# |
| 9 | 8 | 3adantr1 1158 |
. . . . . 6
#
# |
| 10 | 9 | 3adantl1 1155 |
. . . . 5
#
#
|
| 11 | mulap0 8675 |
. . . . . . 7
#
# #
| |
| 12 | 11 | 3adantr1 1158 |
. . . . . 6
#
# # |
| 13 | 12 | 3adantl1 1155 |
. . . . 5
#
#
# |
| 14 | divcanap3 8719 |
. . . . 5
#
| |
| 15 | 6, 10, 13, 14 | syl3anc 1249 |
. . . 4
#
# |
| 16 | simp2 1000 |
. . . . . . . 8
# | |
| 17 | 16, 3 | jca 306 |
. . . . . . 7
#
|
| 18 | simp2 1000 |
. . . . . . . 8
# | |
| 19 | 18, 4 | jca 306 |
. . . . . . 7
#
|
| 20 | mul4 8153 |
. . . . . . 7
| |
| 21 | 17, 19, 20 | syl2an 289 |
. . . . . 6
#
#
|
| 22 | divcanap2 8701 |
. . . . . . 7
#
| |
| 23 | divcanap2 8701 |
. . . . . . 7
#
| |
| 24 | 22, 23 | oveqan12d 5938 |
. . . . . 6
#
#
|
| 25 | 21, 24 | eqtr3d 2228 |
. . . . 5
#
# |
| 26 | 25 | oveq1d 5934 |
. . . 4
#
# |
| 27 | 15, 26 | eqtr3d 2228 |
. . 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 980
wceq 1364
wcel 2164
class class class wbr 4030 (class class class)co 5919
cc 7872
cc0 7874
cmul 7879
# cap 8602 cdiv 8693 |
| 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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-br 4031 df-opab 4092 df-id 4325 df-po 4328 df-iso 4329 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-iota 5216 df-fun 5257 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 |
| This theorem is referenced by: divdivdivap 8734 divcanap5 8735 divmul13ap 8736 divmul24ap 8737 divmuldivapi 8793 divmuldivapd 8853 qmulcl 9705 mulexpzap 10653 expaddzap 10657 sqdivap 10677 |
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