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Mirrors > Home > HSE Home > Th. List > orthin | Structured version Visualization version GIF version |
Description: The intersection of orthogonal subspaces is the zero subspace. (Contributed by NM, 24-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
orthin | ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ⊆ (⊥‘𝐵) → (𝐴 ∩ 𝐵) = 0ℋ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrin 4167 | . . . . . 6 ⊢ (𝐴 ⊆ (⊥‘𝐵) → (𝐴 ∩ 𝐵) ⊆ ((⊥‘𝐵) ∩ 𝐵)) | |
2 | incom 4135 | . . . . . 6 ⊢ ((⊥‘𝐵) ∩ 𝐵) = (𝐵 ∩ (⊥‘𝐵)) | |
3 | 1, 2 | sseqtrdi 3971 | . . . . 5 ⊢ (𝐴 ⊆ (⊥‘𝐵) → (𝐴 ∩ 𝐵) ⊆ (𝐵 ∩ (⊥‘𝐵))) |
4 | ocin 29658 | . . . . . 6 ⊢ (𝐵 ∈ Sℋ → (𝐵 ∩ (⊥‘𝐵)) = 0ℋ) | |
5 | 4 | sseq2d 3953 | . . . . 5 ⊢ (𝐵 ∈ Sℋ → ((𝐴 ∩ 𝐵) ⊆ (𝐵 ∩ (⊥‘𝐵)) ↔ (𝐴 ∩ 𝐵) ⊆ 0ℋ)) |
6 | 3, 5 | syl5ib 243 | . . . 4 ⊢ (𝐵 ∈ Sℋ → (𝐴 ⊆ (⊥‘𝐵) → (𝐴 ∩ 𝐵) ⊆ 0ℋ)) |
7 | 6 | adantl 482 | . . 3 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ⊆ (⊥‘𝐵) → (𝐴 ∩ 𝐵) ⊆ 0ℋ)) |
8 | shincl 29743 | . . . 4 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ∩ 𝐵) ∈ Sℋ ) | |
9 | sh0le 29802 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) ∈ Sℋ → 0ℋ ⊆ (𝐴 ∩ 𝐵)) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → 0ℋ ⊆ (𝐴 ∩ 𝐵)) |
11 | 7, 10 | jctird 527 | . 2 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ⊆ (⊥‘𝐵) → ((𝐴 ∩ 𝐵) ⊆ 0ℋ ∧ 0ℋ ⊆ (𝐴 ∩ 𝐵)))) |
12 | eqss 3936 | . 2 ⊢ ((𝐴 ∩ 𝐵) = 0ℋ ↔ ((𝐴 ∩ 𝐵) ⊆ 0ℋ ∧ 0ℋ ⊆ (𝐴 ∩ 𝐵))) | |
13 | 11, 12 | syl6ibr 251 | 1 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → (𝐴 ⊆ (⊥‘𝐵) → (𝐴 ∩ 𝐵) = 0ℋ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∩ cin 3886 ⊆ wss 3887 ‘cfv 6433 Sℋ csh 29290 ⊥cort 29292 0ℋc0h 29297 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-hilex 29361 ax-hfvadd 29362 ax-hv0cl 29365 ax-hfvmul 29367 ax-hvmul0 29372 ax-hfi 29441 ax-his2 29445 ax-his3 29446 ax-his4 29447 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-ltxr 11014 df-nn 11974 df-hlim 29334 df-sh 29569 df-ch 29583 df-oc 29614 df-ch0 29615 |
This theorem is referenced by: atomli 30744 chirredlem3 30754 |
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