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Theorem orthcom 28888

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 6673	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (∗‘(𝐴 ·_ih 𝐵)) = (∗‘0))
2		cj0 14520	. . . 4 ⊢ (∗‘0) = 0
3	1, 2	syl6eq 2875	. . 3 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (∗‘(𝐴 ·_ih 𝐵)) = 0)
4		ax-his1 28862	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐵 ·_ih 𝐴) = (∗‘(𝐴 ·_ih 𝐵)))
5	4	ancoms 461	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐵 ·_ih 𝐴) = (∗‘(𝐴 ·_ih 𝐵)))
6	5	eqeq1d 2826	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐵 ·_ih 𝐴) = 0 ↔ (∗‘(𝐴 ·_ih 𝐵)) = 0))
7	3, 6	syl5ibr 248	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ih 𝐵) = 0 → (𝐵 ·_ih 𝐴) = 0))
8		fveq2 6673	. . . 4 ⊢ ((𝐵 ·_ih 𝐴) = 0 → (∗‘(𝐵 ·_ih 𝐴)) = (∗‘0))
9	8, 2	syl6eq 2875	. . 3 ⊢ ((𝐵 ·_ih 𝐴) = 0 → (∗‘(𝐵 ·_ih 𝐴)) = 0)
10		ax-his1 28862	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih 𝐵) = (∗‘(𝐵 ·_ih 𝐴)))
11	10	eqeq1d 2826	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ih 𝐵) = 0 ↔ (∗‘(𝐵 ·_ih 𝐴)) = 0))
12	9, 11	syl5ibr 248	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐵 ·_ih 𝐴) = 0 → (𝐴 ·_ih 𝐵) = 0))
13	7, 12	impbid 214	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ih 𝐵) = 0 ↔ (𝐵 ·_ih 𝐴) = 0))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 0cc0 10540 ∗ccj 14458 ℋchba 28699 ·_ih csp 28702

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-his1 28862

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-2 11703 df-cj 14461 df-re 14462 df-im 14463

This theorem is referenced by: normpythi 28922 ocorth 29071 shorth 29075 h1dei 29330 h1de2i 29333 pjspansn 29357

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