![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > spanss | Structured version Visualization version GIF version |
Description: Ordering relationship for the spans of subsets of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
spanss | ⊢ ((𝐵 ⊆ ℋ ∧ 𝐴 ⊆ 𝐵) → (span‘𝐴) ⊆ (span‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sstr2 3858 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝑥 → 𝐴 ⊆ 𝑥)) | |
2 | 1 | ralrimivw 3126 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥 ∈ Sℋ (𝐵 ⊆ 𝑥 → 𝐴 ⊆ 𝑥)) |
3 | ss2rab 3930 | . . . . 5 ⊢ ({𝑥 ∈ Sℋ ∣ 𝐵 ⊆ 𝑥} ⊆ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥} ↔ ∀𝑥 ∈ Sℋ (𝐵 ⊆ 𝑥 → 𝐴 ⊆ 𝑥)) | |
4 | 2, 3 | sylibr 226 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ Sℋ ∣ 𝐵 ⊆ 𝑥} ⊆ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥}) |
5 | intss 4766 | . . . 4 ⊢ ({𝑥 ∈ Sℋ ∣ 𝐵 ⊆ 𝑥} ⊆ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥} → ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥} ⊆ ∩ {𝑥 ∈ Sℋ ∣ 𝐵 ⊆ 𝑥}) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥} ⊆ ∩ {𝑥 ∈ Sℋ ∣ 𝐵 ⊆ 𝑥}) |
7 | 6 | adantl 474 | . 2 ⊢ ((𝐵 ⊆ ℋ ∧ 𝐴 ⊆ 𝐵) → ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥} ⊆ ∩ {𝑥 ∈ Sℋ ∣ 𝐵 ⊆ 𝑥}) |
8 | sstr 3859 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℋ) → 𝐴 ⊆ ℋ) | |
9 | 8 | ancoms 451 | . . 3 ⊢ ((𝐵 ⊆ ℋ ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ ℋ) |
10 | spanval 28906 | . . 3 ⊢ (𝐴 ⊆ ℋ → (span‘𝐴) = ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥}) | |
11 | 9, 10 | syl 17 | . 2 ⊢ ((𝐵 ⊆ ℋ ∧ 𝐴 ⊆ 𝐵) → (span‘𝐴) = ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥}) |
12 | spanval 28906 | . . 3 ⊢ (𝐵 ⊆ ℋ → (span‘𝐵) = ∩ {𝑥 ∈ Sℋ ∣ 𝐵 ⊆ 𝑥}) | |
13 | 12 | adantr 473 | . 2 ⊢ ((𝐵 ⊆ ℋ ∧ 𝐴 ⊆ 𝐵) → (span‘𝐵) = ∩ {𝑥 ∈ Sℋ ∣ 𝐵 ⊆ 𝑥}) |
14 | 7, 11, 13 | 3sstr4d 3897 | 1 ⊢ ((𝐵 ⊆ ℋ ∧ 𝐴 ⊆ 𝐵) → (span‘𝐴) ⊆ (span‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∀wral 3081 {crab 3085 ⊆ wss 3822 ∩ cint 4745 ‘cfv 6185 ℋchba 28490 Sℋ csh 28499 spancspn 28503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-1cn 10391 ax-addcl 10393 ax-hilex 28570 ax-hfvadd 28571 ax-hv0cl 28574 ax-hfvmul 28576 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-map 8206 df-nn 11438 df-hlim 28543 df-sh 28778 df-ch 28792 df-span 28882 |
This theorem is referenced by: spanssoc 28922 span0 29115 spanuni 29117 spansnpji 29151 shatomistici 29934 |
Copyright terms: Public domain | W3C validator |