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Mirrors > Home > HSE Home > Th. List > shorth | Structured version Visualization version GIF version |
Description: Members of orthogonal subspaces are orthogonal. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shorth | ⊢ (𝐻 ∈ Sℋ → (𝐺 ⊆ (⊥‘𝐻) → ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻) → (𝐴 ·ih 𝐵) = 0))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3968 | . . . . . 6 ⊢ (𝐺 ⊆ (⊥‘𝐻) → (𝐴 ∈ 𝐺 → 𝐴 ∈ (⊥‘𝐻))) | |
2 | 1 | anim1d 610 | . . . . 5 ⊢ (𝐺 ⊆ (⊥‘𝐻) → ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻) → (𝐴 ∈ (⊥‘𝐻) ∧ 𝐵 ∈ 𝐻))) |
3 | 2 | imp 406 | . . . 4 ⊢ ((𝐺 ⊆ (⊥‘𝐻) ∧ (𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻)) → (𝐴 ∈ (⊥‘𝐻) ∧ 𝐵 ∈ 𝐻)) |
4 | 3 | ancomd 461 | . . 3 ⊢ ((𝐺 ⊆ (⊥‘𝐻) ∧ (𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻)) → (𝐵 ∈ 𝐻 ∧ 𝐴 ∈ (⊥‘𝐻))) |
5 | shocorth 31017 | . . . . 5 ⊢ (𝐻 ∈ Sℋ → ((𝐵 ∈ 𝐻 ∧ 𝐴 ∈ (⊥‘𝐻)) → (𝐵 ·ih 𝐴) = 0)) | |
6 | 5 | imp 406 | . . . 4 ⊢ ((𝐻 ∈ Sℋ ∧ (𝐵 ∈ 𝐻 ∧ 𝐴 ∈ (⊥‘𝐻))) → (𝐵 ·ih 𝐴) = 0) |
7 | shss 30935 | . . . . . . . 8 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ) | |
8 | 7 | sseld 3974 | . . . . . . 7 ⊢ (𝐻 ∈ Sℋ → (𝐵 ∈ 𝐻 → 𝐵 ∈ ℋ)) |
9 | shocss 31011 | . . . . . . . 8 ⊢ (𝐻 ∈ Sℋ → (⊥‘𝐻) ⊆ ℋ) | |
10 | 9 | sseld 3974 | . . . . . . 7 ⊢ (𝐻 ∈ Sℋ → (𝐴 ∈ (⊥‘𝐻) → 𝐴 ∈ ℋ)) |
11 | 8, 10 | anim12d 608 | . . . . . 6 ⊢ (𝐻 ∈ Sℋ → ((𝐵 ∈ 𝐻 ∧ 𝐴 ∈ (⊥‘𝐻)) → (𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ))) |
12 | 11 | imp 406 | . . . . 5 ⊢ ((𝐻 ∈ Sℋ ∧ (𝐵 ∈ 𝐻 ∧ 𝐴 ∈ (⊥‘𝐻))) → (𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ)) |
13 | orthcom 30833 | . . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝐵 ·ih 𝐴) = 0 ↔ (𝐴 ·ih 𝐵) = 0)) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ ((𝐻 ∈ Sℋ ∧ (𝐵 ∈ 𝐻 ∧ 𝐴 ∈ (⊥‘𝐻))) → ((𝐵 ·ih 𝐴) = 0 ↔ (𝐴 ·ih 𝐵) = 0)) |
15 | 6, 14 | mpbid 231 | . . 3 ⊢ ((𝐻 ∈ Sℋ ∧ (𝐵 ∈ 𝐻 ∧ 𝐴 ∈ (⊥‘𝐻))) → (𝐴 ·ih 𝐵) = 0) |
16 | 4, 15 | sylan2 592 | . 2 ⊢ ((𝐻 ∈ Sℋ ∧ (𝐺 ⊆ (⊥‘𝐻) ∧ (𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻))) → (𝐴 ·ih 𝐵) = 0) |
17 | 16 | exp32 420 | 1 ⊢ (𝐻 ∈ Sℋ → (𝐺 ⊆ (⊥‘𝐻) → ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻) → (𝐴 ·ih 𝐵) = 0))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⊆ wss 3941 ‘cfv 6534 (class class class)co 7402 0cc0 11107 ℋchba 30644 ·ih csp 30647 Sℋ csh 30653 ⊥cort 30655 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-hilex 30724 ax-hfvadd 30725 ax-hv0cl 30728 ax-hfvmul 30730 ax-hvmul0 30735 ax-hfi 30804 ax-his1 30807 ax-his2 30808 ax-his3 30809 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-po 5579 df-so 5580 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-2 12273 df-cj 15044 df-re 15045 df-im 15046 df-sh 30932 df-oc 30977 |
This theorem is referenced by: pjoi0 31442 |
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