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Theorem orthcom 30628

Description: Orthogonality commutes. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)

**Assertion**
Ref	Expression
orthcom	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ih 𝐵) = 0 ↔ (𝐵 ·_ih 𝐴) = 0))

**Proof of Theorem orthcom**
Step	Hyp	Ref	Expression
1		fveq2 6890	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (∗‘(𝐴 ·_ih 𝐵)) = (∗‘0))
2		cj0 15109	. . . 4 ⊢ (∗‘0) = 0
3	1, 2	eqtrdi 2786	. . 3 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (∗‘(𝐴 ·_ih 𝐵)) = 0)
4		ax-his1 30602	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐵 ·_ih 𝐴) = (∗‘(𝐴 ·_ih 𝐵)))
5	4	ancoms 457	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐵 ·_ih 𝐴) = (∗‘(𝐴 ·_ih 𝐵)))
6	5	eqeq1d 2732	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐵 ·_ih 𝐴) = 0 ↔ (∗‘(𝐴 ·_ih 𝐵)) = 0))
7	3, 6	imbitrrid 245	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ih 𝐵) = 0 → (𝐵 ·_ih 𝐴) = 0))
8		fveq2 6890	. . . 4 ⊢ ((𝐵 ·_ih 𝐴) = 0 → (∗‘(𝐵 ·_ih 𝐴)) = (∗‘0))
9	8, 2	eqtrdi 2786	. . 3 ⊢ ((𝐵 ·_ih 𝐴) = 0 → (∗‘(𝐵 ·_ih 𝐴)) = 0)
10		ax-his1 30602	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih 𝐵) = (∗‘(𝐵 ·_ih 𝐴)))
11	10	eqeq1d 2732	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ih 𝐵) = 0 ↔ (∗‘(𝐵 ·_ih 𝐴)) = 0))
12	9, 11	imbitrrid 245	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐵 ·_ih 𝐴) = 0 → (𝐴 ·_ih 𝐵) = 0))
13	7, 12	impbid 211	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ih 𝐵) = 0 ↔ (𝐵 ·_ih 𝐴) = 0))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ‘cfv 6542 (class class class)co 7411 0cc0 11112 ∗ccj 15047 ℋchba 30439 ·_ih csp 30442

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 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-his1 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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-2 12279 df-cj 15050 df-re 15051 df-im 15052

This theorem is referenced by: normpythi 30662 ocorth 30811 shorth 30815 h1dei 31070 h1de2i 31073 pjspansn 31097

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