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Mirrors > Home > MPE Home > Th. List > Mathboxes > pexmidALTN | Structured version Visualization version GIF version |
Description: Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 37104. Lemma 3.3(2) in [Holland95] p. 215. In our proof, we use the variables 𝑋, 𝑀, 𝑝, 𝑞, 𝑟 in place of Hollands' l, m, P, Q, L respectively. TODO: should we make this obsolete? (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pexmidALT.a | ⊢ 𝐴 = (Atoms‘𝐾) |
pexmidALT.p | ⊢ + = (+𝑃‘𝐾) |
pexmidALT.o | ⊢ ⊥ = (⊥𝑃‘𝐾) |
Ref | Expression |
---|---|
pexmidALTN | ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝑋 = ∅ → 𝑋 = ∅) | |
2 | fveq2 6670 | . . . 4 ⊢ (𝑋 = ∅ → ( ⊥ ‘𝑋) = ( ⊥ ‘∅)) | |
3 | 1, 2 | oveq12d 7174 | . . 3 ⊢ (𝑋 = ∅ → (𝑋 + ( ⊥ ‘𝑋)) = (∅ + ( ⊥ ‘∅))) |
4 | pexmidALT.a | . . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | pexmidALT.o | . . . . . . . 8 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
6 | 4, 5 | pol0N 37060 | . . . . . . 7 ⊢ (𝐾 ∈ HL → ( ⊥ ‘∅) = 𝐴) |
7 | eqimss 4023 | . . . . . . 7 ⊢ (( ⊥ ‘∅) = 𝐴 → ( ⊥ ‘∅) ⊆ 𝐴) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝐾 ∈ HL → ( ⊥ ‘∅) ⊆ 𝐴) |
9 | pexmidALT.p | . . . . . . 7 ⊢ + = (+𝑃‘𝐾) | |
10 | 4, 9 | padd02 36963 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘∅) ⊆ 𝐴) → (∅ + ( ⊥ ‘∅)) = ( ⊥ ‘∅)) |
11 | 8, 10 | mpdan 685 | . . . . 5 ⊢ (𝐾 ∈ HL → (∅ + ( ⊥ ‘∅)) = ( ⊥ ‘∅)) |
12 | 11, 6 | eqtrd 2856 | . . . 4 ⊢ (𝐾 ∈ HL → (∅ + ( ⊥ ‘∅)) = 𝐴) |
13 | 12 | ad2antrr 724 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (∅ + ( ⊥ ‘∅)) = 𝐴) |
14 | 3, 13 | sylan9eqr 2878 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) ∧ 𝑋 = ∅) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴) |
15 | 4, 9, 5 | pexmidlem8N 37128 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅)) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴) |
16 | 15 | anassrs 470 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) ∧ 𝑋 ≠ ∅) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴) |
17 | 14, 16 | pm2.61dane 3104 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ⊆ wss 3936 ∅c0 4291 ‘cfv 6355 (class class class)co 7156 Atomscatm 36414 HLchlt 36501 +𝑃cpadd 36946 ⊥𝑃cpolN 37053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-riotaBAD 36104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-undef 7939 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-clat 17718 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-psubsp 36654 df-pmap 36655 df-padd 36947 df-polarityN 37054 df-psubclN 37086 |
This theorem is referenced by: (None) |
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