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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 31042 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ 𝐶) ∈ ℋ) | |
2 | ax-his1 31111 | . . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ·ℎ 𝐶) ∈ ℋ) → (𝐵 ·ih (𝐴 ·ℎ 𝐶)) = (∗‘((𝐴 ·ℎ 𝐶) ·ih 𝐵))) | |
3 | 1, 2 | sylan2 593 | . . . 4 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ)) → (𝐵 ·ih (𝐴 ·ℎ 𝐶)) = (∗‘((𝐴 ·ℎ 𝐶) ·ih 𝐵))) |
4 | 3 | 3impb 1114 | . . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 ·ih (𝐴 ·ℎ 𝐶)) = (∗‘((𝐴 ·ℎ 𝐶) ·ih 𝐵))) |
5 | 4 | 3com12 1122 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·ih (𝐴 ·ℎ 𝐶)) = (∗‘((𝐴 ·ℎ 𝐶) ·ih 𝐵))) |
6 | ax-his3 31113 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ℎ 𝐶) ·ih 𝐵) = (𝐴 · (𝐶 ·ih 𝐵))) | |
7 | 6 | 3com23 1125 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·ℎ 𝐶) ·ih 𝐵) = (𝐴 · (𝐶 ·ih 𝐵))) |
8 | 7 | fveq2d 6911 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘((𝐴 ·ℎ 𝐶) ·ih 𝐵)) = (∗‘(𝐴 · (𝐶 ·ih 𝐵)))) |
9 | hicl 31109 | . . . . . 6 ⊢ ((𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐶 ·ih 𝐵) ∈ ℂ) | |
10 | cjmul 15178 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝐶 ·ih 𝐵) ∈ ℂ) → (∗‘(𝐴 · (𝐶 ·ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·ih 𝐵)))) | |
11 | 9, 10 | sylan2 593 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ)) → (∗‘(𝐴 · (𝐶 ·ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·ih 𝐵)))) |
12 | 11 | 3impb 1114 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∗‘(𝐴 · (𝐶 ·ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·ih 𝐵)))) |
13 | 12 | 3com23 1125 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘(𝐴 · (𝐶 ·ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·ih 𝐵)))) |
14 | ax-his1 31111 | . . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·ih 𝐶) = (∗‘(𝐶 ·ih 𝐵))) | |
15 | 14 | 3adant1 1129 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·ih 𝐶) = (∗‘(𝐶 ·ih 𝐵))) |
16 | 15 | oveq2d 7447 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((∗‘𝐴) · (𝐵 ·ih 𝐶)) = ((∗‘𝐴) · (∗‘(𝐶 ·ih 𝐵)))) |
17 | 13, 16 | eqtr4d 2778 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘(𝐴 · (𝐶 ·ih 𝐵))) = ((∗‘𝐴) · (𝐵 ·ih 𝐶))) |
18 | 5, 8, 17 | 3eqtrd 2779 | 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 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 · cmul 11158 ∗ccj 15132 ℋchba 30948 ·ℎ csm 30950 ·ih csp 30951 |
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-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-hfvmul 31034 ax-hfi 31108 ax-his1 31111 ax-his3 31113 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-2 12327 df-cj 15135 df-re 15136 df-im 15137 |
This theorem is referenced by: his52 31116 his35 31117 normlem0 31138 normlem9 31147 bcseqi 31149 polid2i 31186 pjhthlem1 31420 eigrei 31863 eigposi 31865 eigorthi 31866 brafnmul 31980 lnopunilem1 32039 hmopm 32050 cnlnadjlem6 32101 adjlnop 32115 |
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