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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 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
joinlmuladdmuld.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
joinlmuladdmuld.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
joinlmuladdmuld.4 | ⊢ (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷) |
Ref | Expression |
---|---|
joinlmuladdmuld | ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | joinlmuladdmuld.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | joinlmuladdmuld.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
3 | joinlmuladdmuld.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | 1, 2, 3 | adddird 11271 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = ((𝐴 · 𝐵) + (𝐶 · 𝐵))) |
5 | joinlmuladdmuld.4 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷) | |
6 | 4, 5 | eqtrd 2765 | 1 ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 (class class class)co 7419 ℂcc 11138 + caddc 11143 · cmul 11145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-addcl 11200 ax-mulcom 11204 ax-distr 11207 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-iota 6501 df-fv 6557 df-ov 7422 |
This theorem is referenced by: 1p1times 11417 div4p1lem1div2 12500 ltdifltdiv 13835 discr1 14237 arisum 15842 bezoutlem3 16520 bezoutlem4 16521 mbfi1fseqlem4 25692 itgmulc2 25807 tangtx 26485 binom4 26827 axcontlem8 28854 zringfrac 33369 int-rightdistd 43752 fmtnorec2lem 47019 joinlmuladdmuli 48392 |
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