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| Mirrors > Home > ILE Home > Th. List > divmulassap | Unicode version | ||
| Description: An associative law for division and multiplication. (Contributed by Jim Kingdon, 24-Jan-2022.) |
| Ref | Expression |
|---|---|
| divmulassap |
     ![/\](wedge.gif "/\"){kind=small}  
  #      |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpl1 1003 |
. . . . 5
# | |
| 2 | simpl2 1004 |
. . . . 5
# | |
| 3 | simpr 110 |
. . . . 5
# # | |
| 4 | divassap 8783 |
. . . . 5
# | |
| 5 | 1, 2, 3, 4 | syl3anc 1250 |
. . . 4
# |
| 6 | 5 | eqcomd 2212 |
. . 3
#
|
| 7 | 6 | oveq1d 5972 |
. 2
# |
| 8 | mulcl 8072 |
. . . . 5
| |
| 9 | 8 | 3adant3 1020 |
. . . 4
|
| 10 | 9 | adantr 276 |
. . 3
# |
| 11 | simpl3 1005 |
. . 3
# | |
| 12 | div32ap 8785 |
. . 3
#
| |
| 13 | 10, 3, 11, 12 | syl3anc 1250 |
. 2
# |
| 14 | 7, 13 | eqtrd 2239 |
1
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
w3a 981
wceq 1373
wcel 2177
class class class wbr 4051 (class class class)co 5957
cc 7943
cc0 7945
cmul 7950
# cap 8674 cdiv 8765 |
| 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 2179 ax-14 2180 ax-ext 2188 ax-sep 4170 ax-pow 4226 ax-pr 4261 ax-un 4488 ax-setind 4593 ax-cnex 8036 ax-resscn 8037 ax-1cn 8038 ax-1re 8039 ax-icn 8040 ax-addcl 8041 ax-addrcl 8042 ax-mulcl 8043 ax-mulrcl 8044 ax-addcom 8045 ax-mulcom 8046 ax-addass 8047 ax-mulass 8048 ax-distr 8049 ax-i2m1 8050 ax-0lt1 8051 ax-1rid 8052 ax-0id 8053 ax-rnegex 8054 ax-precex 8055 ax-cnre 8056 ax-pre-ltirr 8057 ax-pre-ltwlin 8058 ax-pre-lttrn 8059 ax-pre-apti 8060 ax-pre-ltadd 8061 ax-pre-mulgt0 8062 ax-pre-mulext 8063 |
| 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 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3003 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3857 df-br 4052 df-opab 4114 df-id 4348 df-po 4351 df-iso 4352 df-xp 4689 df-rel 4690 df-cnv 4691 df-co 4692 df-dm 4693 df-iota 5241 df-fun 5282 df-fv 5288 df-riota 5912 df-ov 5960 df-oprab 5961 df-mpo 5962 df-pnf 8129 df-mnf 8130 df-xr 8131 df-ltxr 8132 df-le 8133 df-sub 8265 df-neg 8266 df-reap 8668 df-ap 8675 df-div 8766 |
| This theorem is referenced by: divmulasscomap 8789 |
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