Mathbox for David A. Wheeler |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > joinlmuladdmuli | 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 |
---|---|
joinlmuladdmuli.1 | ⊢ 𝐴 ∈ ℂ |
joinlmuladdmuli.2 | ⊢ 𝐵 ∈ ℂ |
joinlmuladdmuli.3 | ⊢ 𝐶 ∈ ℂ |
joinlmuladdmuli.4 | ⊢ ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷 |
Ref | Expression |
---|---|
joinlmuladdmuli | ⊢ ((𝐴 + 𝐶) · 𝐵) = 𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | joinlmuladdmuli.1 | . . . 4 ⊢ 𝐴 ∈ ℂ | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → 𝐴 ∈ ℂ) |
3 | joinlmuladdmuli.2 | . . . 4 ⊢ 𝐵 ∈ ℂ | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → 𝐵 ∈ ℂ) |
5 | joinlmuladdmuli.3 | . . . 4 ⊢ 𝐶 ∈ ℂ | |
6 | 5 | a1i 11 | . . 3 ⊢ (⊤ → 𝐶 ∈ ℂ) |
7 | joinlmuladdmuli.4 | . . . 4 ⊢ ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷 | |
8 | 7 | a1i 11 | . . 3 ⊢ (⊤ → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷) |
9 | 2, 4, 6, 8 | joinlmuladdmuld 10670 | . 2 ⊢ (⊤ → ((𝐴 + 𝐶) · 𝐵) = 𝐷) |
10 | 9 | mptru 1544 | 1 ⊢ ((𝐴 + 𝐶) · 𝐵) = 𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 (class class class)co 7158 ℂcc 10537 + caddc 10542 · cmul 10544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-addcl 10599 ax-mulcom 10603 ax-distr 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |