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Mirrors > Home > HSE Home > Th. List > his5 | Structured version Visualization version GIF version |
Description: Associative law for inner product. Lemma 3.1(S5) of [Beran] p. 95. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
his5 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·ih (𝐴 ·ℎ 𝐶)) = ((∗‘𝐴) · (𝐵 ·ih 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvmulcl 28210 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ 𝐶) ∈ ℋ) | |
2 | ax-his1 28279 | . . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ·ℎ 𝐶) ∈ ℋ) → (𝐵 ·ih (𝐴 ·ℎ 𝐶)) = (∗‘((𝐴 ·ℎ 𝐶) ·ih 𝐵))) | |
3 | 1, 2 | sylan2 580 | . . . 4 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ)) → (𝐵 ·ih (𝐴 ·ℎ 𝐶)) = (∗‘((𝐴 ·ℎ 𝐶) ·ih 𝐵))) |
4 | 3 | 3impb 1107 | . . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 ·ih (𝐴 ·ℎ 𝐶)) = (∗‘((𝐴 ·ℎ 𝐶) ·ih 𝐵))) |
5 | 4 | 3com12 1117 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·ih (𝐴 ·ℎ 𝐶)) = (∗‘((𝐴 ·ℎ 𝐶) ·ih 𝐵))) |
6 | ax-his3 28281 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ℎ 𝐶) ·ih 𝐵) = (𝐴 · (𝐶 ·ih 𝐵))) | |
7 | 6 | 3com23 1120 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·ℎ 𝐶) ·ih 𝐵) = (𝐴 · (𝐶 ·ih 𝐵))) |
8 | 7 | fveq2d 6336 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘((𝐴 ·ℎ 𝐶) ·ih 𝐵)) = (∗‘(𝐴 · (𝐶 ·ih 𝐵)))) |
9 | hicl 28277 | . . . . . 6 ⊢ ((𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐶 ·ih 𝐵) ∈ ℂ) | |
10 | cjmul 14090 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝐶 ·ih 𝐵) ∈ ℂ) → (∗‘(𝐴 · (𝐶 ·ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·ih 𝐵)))) | |
11 | 9, 10 | sylan2 580 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ)) → (∗‘(𝐴 · (𝐶 ·ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·ih 𝐵)))) |
12 | 11 | 3impb 1107 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∗‘(𝐴 · (𝐶 ·ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·ih 𝐵)))) |
13 | 12 | 3com23 1120 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘(𝐴 · (𝐶 ·ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·ih 𝐵)))) |
14 | ax-his1 28279 | . . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·ih 𝐶) = (∗‘(𝐶 ·ih 𝐵))) | |
15 | 14 | 3adant1 1124 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·ih 𝐶) = (∗‘(𝐶 ·ih 𝐵))) |
16 | 15 | oveq2d 6809 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((∗‘𝐴) · (𝐵 ·ih 𝐶)) = ((∗‘𝐴) · (∗‘(𝐶 ·ih 𝐵)))) |
17 | 13, 16 | eqtr4d 2808 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘(𝐴 · (𝐶 ·ih 𝐵))) = ((∗‘𝐴) · (𝐵 ·ih 𝐶))) |
18 | 5, 8, 17 | 3eqtrd 2809 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·ih (𝐴 ·ℎ 𝐶)) = ((∗‘𝐴) · (𝐵 ·ih 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ‘cfv 6031 (class class class)co 6793 ℂcc 10136 · cmul 10143 ∗ccj 14044 ℋchil 28116 ·ℎ csm 28118 ·ih csp 28119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-hfvmul 28202 ax-hfi 28276 ax-his1 28279 ax-his3 28281 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-2 11281 df-cj 14047 df-re 14048 df-im 14049 |
This theorem is referenced by: his52 28284 his35 28285 normlem0 28306 normlem9 28315 bcseqi 28317 polid2i 28354 pjhthlem1 28590 eigrei 29033 eigposi 29035 eigorthi 29036 brafnmul 29150 lnopunilem1 29209 hmopm 29220 cnlnadjlem6 29271 adjlnop 29285 |
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