his35 - Hilbert Space Explorer

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Theorem his35 30606

Description: Move scalar multiplication to outside of inner product. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

**Assertion**
Ref	Expression
his35	⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ·_ℎ 𝐶) ·_ih (𝐵 ·_ℎ 𝐷)) = ((𝐴 · (∗‘𝐵)) · (𝐶 ·_ih 𝐷)))

**Proof of Theorem his35**
Step	Hyp	Ref	Expression
1		his5 30604	. . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐶 ·_ih (𝐵 ·_ℎ 𝐷)) = ((∗‘𝐵) · (𝐶 ·_ih 𝐷)))
2	1	3expb 1118	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (𝐶 ·_ih (𝐵 ·_ℎ 𝐷)) = ((∗‘𝐵) · (𝐶 ·_ih 𝐷)))
3	2	adantll 710	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (𝐶 ·_ih (𝐵 ·_ℎ 𝐷)) = ((∗‘𝐵) · (𝐶 ·_ih 𝐷)))
4	3	oveq2d 7429	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (𝐴 · (𝐶 ·_ih (𝐵 ·_ℎ 𝐷))) = (𝐴 · ((∗‘𝐵) · (𝐶 ·_ih 𝐷))))
5		simpll 763	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → 𝐴 ∈ ℂ)
6		simprl 767	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → 𝐶 ∈ ℋ)
7		hvmulcl 30531	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐷 ∈ ℋ) → (𝐵 ·_ℎ 𝐷) ∈ ℋ)
8	7	ad2ant2l 742	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (𝐵 ·_ℎ 𝐷) ∈ ℋ)
9		ax-his3 30602	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ (𝐵 ·_ℎ 𝐷) ∈ ℋ) → ((𝐴 ·_ℎ 𝐶) ·_ih (𝐵 ·_ℎ 𝐷)) = (𝐴 · (𝐶 ·_ih (𝐵 ·_ℎ 𝐷))))
10	5, 6, 8, 9	syl3anc 1369	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ·_ℎ 𝐶) ·_ih (𝐵 ·_ℎ 𝐷)) = (𝐴 · (𝐶 ·_ih (𝐵 ·_ℎ 𝐷))))
11		cjcl 15058	. . . 4 ⊢ (𝐵 ∈ ℂ → (∗‘𝐵) ∈ ℂ)
12	11	ad2antlr 723	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (∗‘𝐵) ∈ ℂ)
13		hicl 30598	. . . 4 ⊢ ((𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐶 ·_ih 𝐷) ∈ ℂ)
14	13	adantl 480	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (𝐶 ·_ih 𝐷) ∈ ℂ)
15	5, 12, 14	mulassd 11243	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 · (∗‘𝐵)) · (𝐶 ·_ih 𝐷)) = (𝐴 · ((∗‘𝐵) · (𝐶 ·_ih 𝐷))))
16	4, 10, 15	3eqtr4d 2780	1 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ·_ℎ 𝐶) ·_ih (𝐵 ·_ℎ 𝐷)) = ((𝐴 · (∗‘𝐵)) · (𝐶 ·_ih 𝐷)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ‘cfv 6544 (class class class)co 7413 ℂcc 11112 · cmul 11119 ∗ccj 15049 ℋchba 30437 ·_ℎ csm 30439 ·_ih csp 30440

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-hfvmul 30523 ax-hfi 30597 ax-his1 30600 ax-his3 30602

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-div 11878 df-2 12281 df-cj 15052 df-re 15053 df-im 15054

This theorem is referenced by: his35i 30607 pjhthlem1 30909

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