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| Mirrors > Home > HSE Home > Th. List > hiassdi | Structured version Visualization version GIF version | ||
| Description: Distributive/associative law for inner product, useful for linearity proofs. (Contributed by NM, 10-May-2005.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| hiassdi | ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 ·ℎ 𝐵) +ℎ 𝐶) ·ih 𝐷) = ((𝐴 · (𝐵 ·ih 𝐷)) + (𝐶 ·ih 𝐷))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | hvmulcl 31032 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ℎ 𝐵) ∈ ℋ) | |
| 2 | ax-his2 31102 | . . . 4 ⊢ (((𝐴 ·ℎ 𝐵) ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (((𝐴 ·ℎ 𝐵) +ℎ 𝐶) ·ih 𝐷) = (((𝐴 ·ℎ 𝐵) ·ih 𝐷) + (𝐶 ·ih 𝐷))) | |
| 3 | 2 | 3expb 1121 | . . 3 ⊢ (((𝐴 ·ℎ 𝐵) ∈ ℋ ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 ·ℎ 𝐵) +ℎ 𝐶) ·ih 𝐷) = (((𝐴 ·ℎ 𝐵) ·ih 𝐷) + (𝐶 ·ih 𝐷))) | 
| 4 | 1, 3 | sylan 580 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 ·ℎ 𝐵) +ℎ 𝐶) ·ih 𝐷) = (((𝐴 ·ℎ 𝐵) ·ih 𝐷) + (𝐶 ·ih 𝐷))) | 
| 5 | ax-his3 31103 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ) → ((𝐴 ·ℎ 𝐵) ·ih 𝐷) = (𝐴 · (𝐵 ·ih 𝐷))) | |
| 6 | 5 | 3expa 1119 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ 𝐷 ∈ ℋ) → ((𝐴 ·ℎ 𝐵) ·ih 𝐷) = (𝐴 · (𝐵 ·ih 𝐷))) | 
| 7 | 6 | adantrl 716 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ·ℎ 𝐵) ·ih 𝐷) = (𝐴 · (𝐵 ·ih 𝐷))) | 
| 8 | 7 | oveq1d 7446 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 ·ℎ 𝐵) ·ih 𝐷) + (𝐶 ·ih 𝐷)) = ((𝐴 · (𝐵 ·ih 𝐷)) + (𝐶 ·ih 𝐷))) | 
| 9 | 4, 8 | eqtrd 2777 | 1 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 ·ℎ 𝐵) +ℎ 𝐶) ·ih 𝐷) = ((𝐴 · (𝐵 ·ih 𝐷)) + (𝐶 ·ih 𝐷))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 + caddc 11158 · cmul 11160 ℋchba 30938 +ℎ cva 30939 ·ℎ csm 30940 ·ih csp 30941 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-hfvmul 31024 ax-his2 31102 ax-his3 31103 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 | 
| This theorem is referenced by: unoplin 31939 hmoplin 31961 | 
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