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Mirrors > Home > MPE Home > Th. List > mul4i | Structured version Visualization version GIF version |
Description: Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.) |
Ref | Expression |
---|---|
mul.1 | ⊢ 𝐴 ∈ ℂ |
mul.2 | ⊢ 𝐵 ∈ ℂ |
mul.3 | ⊢ 𝐶 ∈ ℂ |
mul4.4 | ⊢ 𝐷 ∈ ℂ |
Ref | Expression |
---|---|
mul4i | ⊢ ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | mul.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
3 | mul.3 | . 2 ⊢ 𝐶 ∈ ℂ | |
4 | mul4.4 | . 2 ⊢ 𝐷 ∈ ℂ | |
5 | mul4 11427 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷))) | |
6 | 1, 2, 3, 4, 5 | mp4an 693 | 1 ⊢ ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 (class class class)co 7431 ℂcc 11151 · cmul 11158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-mulcl 11215 ax-mulcom 11217 ax-mulass 11219 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 |
This theorem is referenced by: faclbnd4lem1 14329 bposlem8 27350 normlem1 31139 dpmul 32880 dpmul4 32881 |
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