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Mirrors > Home > ILE Home > Th. List > divassap | Unicode version |
Description: An associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.) |
Ref | Expression |
---|---|
divassap | # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recclap 7767 | . . 3 # | |
2 | mulass 7104 | . . 3 | |
3 | 1, 2 | syl3an3 1204 | . 2 # |
4 | mulcl 7100 | . . . 4 | |
5 | 4 | 3adant3 958 | . . 3 # |
6 | simp3l 966 | . . 3 # | |
7 | simp3r 967 | . . 3 # # | |
8 | divrecap 7776 | . . 3 # | |
9 | 5, 6, 7, 8 | syl3anc 1169 | . 2 # |
10 | simp2 939 | . . . 4 # | |
11 | divrecap 7776 | . . . 4 # | |
12 | 10, 6, 7, 11 | syl3anc 1169 | . . 3 # |
13 | 12 | oveq2d 5548 | . 2 # |
14 | 3, 9, 13 | 3eqtr4d 2123 | 1 # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 w3a 919 wceq 1284 wcel 1433 class class class wbr 3785 (class class class)co 5532 cc 6979 cc0 6981 c1 6982 cmul 6986 # cap 7681 cdiv 7760 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 |
This theorem is referenced by: div23ap 7779 div32ap 7780 divmulassap 7783 divmulasscomap 7784 divassapzi 7850 divassapi 7856 divassapd 7912 zdivmul 8437 |
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