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Mirrors > Home > MPE Home > Th. List > joinlmuladdmuld | Structured version Visualization version GIF version |
Description: Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.) |
Ref | Expression |
---|---|
joinlmuladdmuld.1 | โข (๐ โ ๐ด โ โ) |
joinlmuladdmuld.2 | โข (๐ โ ๐ต โ โ) |
joinlmuladdmuld.3 | โข (๐ โ ๐ถ โ โ) |
joinlmuladdmuld.4 | โข (๐ โ ((๐ด ยท ๐ต) + (๐ถ ยท ๐ต)) = ๐ท) |
Ref | Expression |
---|---|
joinlmuladdmuld | โข (๐ โ ((๐ด + ๐ถ) ยท ๐ต) = ๐ท) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | joinlmuladdmuld.1 | . . 3 โข (๐ โ ๐ด โ โ) | |
2 | joinlmuladdmuld.3 | . . 3 โข (๐ โ ๐ถ โ โ) | |
3 | joinlmuladdmuld.2 | . . 3 โข (๐ โ ๐ต โ โ) | |
4 | 1, 2, 3 | adddird 11185 | . 2 โข (๐ โ ((๐ด + ๐ถ) ยท ๐ต) = ((๐ด ยท ๐ต) + (๐ถ ยท ๐ต))) |
5 | joinlmuladdmuld.4 | . 2 โข (๐ โ ((๐ด ยท ๐ต) + (๐ถ ยท ๐ต)) = ๐ท) | |
6 | 4, 5 | eqtrd 2773 | 1 โข (๐ โ ((๐ด + ๐ถ) ยท ๐ต) = ๐ท) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 (class class class)co 7358 โcc 11054 + caddc 11059 ยท cmul 11061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-addcl 11116 ax-mulcom 11120 ax-distr 11123 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-ov 7361 |
This theorem is referenced by: 1p1times 11331 div4p1lem1div2 12413 ltdifltdiv 13745 discr1 14148 arisum 15750 bezoutlem3 16427 bezoutlem4 16428 mbfi1fseqlem4 25099 itgmulc2 25214 tangtx 25878 binom4 26216 axcontlem8 27962 int-rightdistd 42541 fmtnorec2lem 45820 joinlmuladdmuli 47306 |
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