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Theorem orthcom 28887

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 6672	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (∗‘(𝐴 ·_ih 𝐵)) = (∗‘0))
2		cj0 14519	. . . 4 ⊢ (∗‘0) = 0
3	1, 2	syl6eq 2874	. . 3 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (∗‘(𝐴 ·_ih 𝐵)) = 0)
4		ax-his1 28861	. . . . 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 2825	. . 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 6672	. . . 4 ⊢ ((𝐵 ·_ih 𝐴) = 0 → (∗‘(𝐵 ·_ih 𝐴)) = (∗‘0))
9	8, 2	syl6eq 2874	. . 3 ⊢ ((𝐵 ·_ih 𝐴) = 0 → (∗‘(𝐵 ·_ih 𝐴)) = 0)
10		ax-his1 28861	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih 𝐵) = (∗‘(𝐵 ·_ih 𝐴)))
11	10	eqeq1d 2825	. . 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 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 0cc0 10539 ∗ccj 14457 ℋchba 28698 ·_ih csp 28701

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-his1 28861

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-2 11703 df-cj 14460 df-re 14461 df-im 14462

This theorem is referenced by: normpythi 28921 ocorth 29070 shorth 29074 h1dei 29329 h1de2i 29332 pjspansn 29356

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