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Theorem orthcom 28305

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 6333	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (∗‘(𝐴 ·_ih 𝐵)) = (∗‘0))
2		cj0 14106	. . . 4 ⊢ (∗‘0) = 0
3	1, 2	syl6eq 2821	. . 3 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (∗‘(𝐴 ·_ih 𝐵)) = 0)
4		ax-his1 28279	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐵 ·_ih 𝐴) = (∗‘(𝐴 ·_ih 𝐵)))
5	4	ancoms 446	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐵 ·_ih 𝐴) = (∗‘(𝐴 ·_ih 𝐵)))
6	5	eqeq1d 2773	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐵 ·_ih 𝐴) = 0 ↔ (∗‘(𝐴 ·_ih 𝐵)) = 0))
7	3, 6	syl5ibr 236	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ih 𝐵) = 0 → (𝐵 ·_ih 𝐴) = 0))
8		fveq2 6333	. . . 4 ⊢ ((𝐵 ·_ih 𝐴) = 0 → (∗‘(𝐵 ·_ih 𝐴)) = (∗‘0))
9	8, 2	syl6eq 2821	. . 3 ⊢ ((𝐵 ·_ih 𝐴) = 0 → (∗‘(𝐵 ·_ih 𝐴)) = 0)
10		ax-his1 28279	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih 𝐵) = (∗‘(𝐵 ·_ih 𝐴)))
11	10	eqeq1d 2773	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ih 𝐵) = 0 ↔ (∗‘(𝐵 ·_ih 𝐴)) = 0))
12	9, 11	syl5ibr 236	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐵 ·_ih 𝐴) = 0 → (𝐴 ·_ih 𝐵) = 0))
13	7, 12	impbid 202	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ih 𝐵) = 0 ↔ (𝐵 ·_ih 𝐴) = 0))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ‘cfv 6030 (class class class)co 6796 0cc0 10142 ∗ccj 14044 ℋchil 28116 ·_ih csp 28119

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-his1 28279

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-po 5171 df-so 5172 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-er 7900 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-2 11285 df-cj 14047 df-re 14048 df-im 14049

This theorem is referenced by: normpythi 28339 ocorth 28490 shorth 28494 h1dei 28749 h1de2i 28752 pjspansn 28776

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