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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 11261 | . 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 2767 | 1 โข (๐ โ ((๐ด + ๐ถ) ยท ๐ต) = ๐ท) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1534 โ wcel 2099 (class class class)co 7414 โcc 11128 + caddc 11133 ยท cmul 11135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-addcl 11190 ax-mulcom 11194 ax-distr 11197 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-iota 6494 df-fv 6550 df-ov 7417 |
This theorem is referenced by: 1p1times 11407 div4p1lem1div2 12489 ltdifltdiv 13823 discr1 14225 arisum 15830 bezoutlem3 16508 bezoutlem4 16509 mbfi1fseqlem4 25635 itgmulc2 25750 tangtx 26427 binom4 26769 axcontlem8 28769 int-rightdistd 43533 fmtnorec2lem 46805 joinlmuladdmuli 48129 |
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