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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 34257. 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 6103 | . . . 4 ⊢ (𝑋 = ∅ → ( ⊥ ‘𝑋) = ( ⊥ ‘∅)) | |
3 | 1, 2 | oveq12d 6567 | . . 3 ⊢ (𝑋 = ∅ → (𝑋 + ( ⊥ ‘𝑋)) = (∅ + ( ⊥ ‘∅))) |
4 | pexmidALT.a | . . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | pexmidALT.o | . . . . . . . 8 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
6 | 4, 5 | pol0N 34213 | . . . . . . 7 ⊢ (𝐾 ∈ HL → ( ⊥ ‘∅) = 𝐴) |
7 | eqimss 3620 | . . . . . . 7 ⊢ (( ⊥ ‘∅) = 𝐴 → ( ⊥ ‘∅) ⊆ 𝐴) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝐾 ∈ HL → ( ⊥ ‘∅) ⊆ 𝐴) |
9 | pexmidALT.p | . . . . . . 7 ⊢ + = (+𝑃‘𝐾) | |
10 | 4, 9 | padd02 34116 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘∅) ⊆ 𝐴) → (∅ + ( ⊥ ‘∅)) = ( ⊥ ‘∅)) |
11 | 8, 10 | mpdan 699 | . . . . 5 ⊢ (𝐾 ∈ HL → (∅ + ( ⊥ ‘∅)) = ( ⊥ ‘∅)) |
12 | 11, 6 | eqtrd 2644 | . . . 4 ⊢ (𝐾 ∈ HL → (∅ + ( ⊥ ‘∅)) = 𝐴) |
13 | 12 | ad2antrr 758 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (∅ + ( ⊥ ‘∅)) = 𝐴) |
14 | 3, 13 | sylan9eqr 2666 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) ∧ 𝑋 = ∅) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴) |
15 | 4, 9, 5 | pexmidlem8N 34281 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅)) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴) |
16 | 15 | anassrs 678 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) ∧ 𝑋 ≠ ∅) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴) |
17 | 14, 16 | pm2.61dane 2869 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ⊆ wss 3540 ∅c0 3874 ‘cfv 5804 (class class class)co 6549 Atomscatm 33568 HLchlt 33655 +𝑃cpadd 34099 ⊥𝑃cpolN 34206 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-riotaBAD 33257 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-undef 7286 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-psubsp 33807 df-pmap 33808 df-padd 34100 df-polarityN 34207 df-psubclN 34239 |
This theorem is referenced by: (None) |
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