Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > div12ap | Unicode version |
Description: A commutative/associative law for division. (Contributed by Jim Kingdon, 25-Feb-2020.) |
Ref | Expression |
---|---|
div12ap | # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divclap 8555 | . . . . 5 # | |
2 | 1 | 3expb 1186 | . . . 4 # |
3 | mulcom 7863 | . . . 4 | |
4 | 2, 3 | sylan2 284 | . . 3 # |
5 | 4 | 3impb 1181 | . 2 # |
6 | div13ap 8570 | . . 3 # | |
7 | 6 | 3comr 1193 | . 2 # |
8 | divclap 8555 | . . . . . 6 # | |
9 | 8 | 3expb 1186 | . . . . 5 # |
10 | mulcom 7863 | . . . . 5 | |
11 | 9, 10 | sylan 281 | . . . 4 # |
12 | 11 | 3impa 1177 | . . 3 # |
13 | 12 | 3com23 1191 | . 2 # |
14 | 5, 7, 13 | 3eqtrd 2194 | 1 # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 963 wceq 1335 wcel 2128 class class class wbr 3967 (class class class)co 5826 cc 7732 cc0 7734 cmul 7739 # cap 8460 cdiv 8549 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-cnex 7825 ax-resscn 7826 ax-1cn 7827 ax-1re 7828 ax-icn 7829 ax-addcl 7830 ax-addrcl 7831 ax-mulcl 7832 ax-mulrcl 7833 ax-addcom 7834 ax-mulcom 7835 ax-addass 7836 ax-mulass 7837 ax-distr 7838 ax-i2m1 7839 ax-0lt1 7840 ax-1rid 7841 ax-0id 7842 ax-rnegex 7843 ax-precex 7844 ax-cnre 7845 ax-pre-ltirr 7846 ax-pre-ltwlin 7847 ax-pre-lttrn 7848 ax-pre-apti 7849 ax-pre-ltadd 7850 ax-pre-mulgt0 7851 ax-pre-mulext 7852 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-id 4255 df-po 4258 df-iso 4259 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-iota 5137 df-fun 5174 df-fv 5180 df-riota 5782 df-ov 5829 df-oprab 5830 df-mpo 5831 df-pnf 7916 df-mnf 7917 df-xr 7918 df-ltxr 7919 df-le 7920 df-sub 8052 df-neg 8053 df-reap 8454 df-ap 8461 df-div 8550 |
This theorem is referenced by: div2negap 8612 div12apd 8704 efival 11640 cos01bnd 11666 cos01gt0 11670 sincosq4sgn 13220 |
Copyright terms: Public domain | W3C validator |