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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 10205 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷))) | |
6 | 1, 2, 3, 4, 5 | mp4an 709 | 1 ⊢ ((𝐴 · 𝐵) · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · (𝐵 · 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 (class class class)co 6650 ℂcc 9934 · cmul 9941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-mulcl 9998 ax-mulcom 10000 ax-mulass 10002 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: faclbnd4lem1 13080 bposlem8 25016 normlem1 27967 dpmul 29621 dpmul4 29622 |
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