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Mirrors > Home > HSE Home > Th. List > shlej1 | Structured version Visualization version GIF version |
Description: Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 22-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shlej1 | ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∨ℋ 𝐶) ⊆ (𝐵 ∨ℋ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 487 | . . 3 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵) | |
2 | unss1 4155 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶)) | |
3 | simpl1 1187 | . . . . . . 7 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ Sℋ ) | |
4 | shss 28981 | . . . . . . 7 ⊢ (𝐴 ∈ Sℋ → 𝐴 ⊆ ℋ) | |
5 | 3, 4 | syl 17 | . . . . . 6 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ ℋ) |
6 | simpl3 1189 | . . . . . . 7 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → 𝐶 ∈ Sℋ ) | |
7 | shss 28981 | . . . . . . 7 ⊢ (𝐶 ∈ Sℋ → 𝐶 ⊆ ℋ) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → 𝐶 ⊆ ℋ) |
9 | 5, 8 | unssd 4162 | . . . . 5 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∪ 𝐶) ⊆ ℋ) |
10 | simpl2 1188 | . . . . . . 7 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ Sℋ ) | |
11 | shss 28981 | . . . . . . 7 ⊢ (𝐵 ∈ Sℋ → 𝐵 ⊆ ℋ) | |
12 | 10, 11 | syl 17 | . . . . . 6 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ⊆ ℋ) |
13 | 12, 8 | unssd 4162 | . . . . 5 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝐵 ∪ 𝐶) ⊆ ℋ) |
14 | occon2 29059 | . . . . 5 ⊢ (((𝐴 ∪ 𝐶) ⊆ ℋ ∧ (𝐵 ∪ 𝐶) ⊆ ℋ) → ((𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶) → (⊥‘(⊥‘(𝐴 ∪ 𝐶))) ⊆ (⊥‘(⊥‘(𝐵 ∪ 𝐶))))) | |
15 | 9, 13, 14 | syl2anc 586 | . . . 4 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → ((𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶) → (⊥‘(⊥‘(𝐴 ∪ 𝐶))) ⊆ (⊥‘(⊥‘(𝐵 ∪ 𝐶))))) |
16 | 2, 15 | syl5 34 | . . 3 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ⊆ 𝐵 → (⊥‘(⊥‘(𝐴 ∪ 𝐶))) ⊆ (⊥‘(⊥‘(𝐵 ∪ 𝐶))))) |
17 | 1, 16 | mpd 15 | . 2 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → (⊥‘(⊥‘(𝐴 ∪ 𝐶))) ⊆ (⊥‘(⊥‘(𝐵 ∪ 𝐶)))) |
18 | shjval 29122 | . . 3 ⊢ ((𝐴 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) → (𝐴 ∨ℋ 𝐶) = (⊥‘(⊥‘(𝐴 ∪ 𝐶)))) | |
19 | 3, 6, 18 | syl2anc 586 | . 2 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∨ℋ 𝐶) = (⊥‘(⊥‘(𝐴 ∪ 𝐶)))) |
20 | shjval 29122 | . . 3 ⊢ ((𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) → (𝐵 ∨ℋ 𝐶) = (⊥‘(⊥‘(𝐵 ∪ 𝐶)))) | |
21 | 10, 6, 20 | syl2anc 586 | . 2 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝐵 ∨ℋ 𝐶) = (⊥‘(⊥‘(𝐵 ∪ 𝐶)))) |
22 | 17, 19, 21 | 3sstr4d 4014 | 1 ⊢ (((𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ∧ 𝐶 ∈ Sℋ ) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∨ℋ 𝐶) ⊆ (𝐵 ∨ℋ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∪ cun 3934 ⊆ wss 3936 ‘cfv 6350 (class class class)co 7150 ℋchba 28690 Sℋ csh 28699 ⊥cort 28701 ∨ℋ chj 28704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-hilex 28770 ax-hfvadd 28771 ax-hv0cl 28774 ax-hfvmul 28776 ax-hvmul0 28781 ax-hfi 28850 ax-his2 28854 ax-his3 28855 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 df-sh 28978 df-oc 29023 df-chj 29081 |
This theorem is referenced by: shlej2 29132 shlej1i 29149 chlej1 29281 |
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