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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 37093. 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 6673 | . . . 4 ⊢ (𝑋 = ∅ → ( ⊥ ‘𝑋) = ( ⊥ ‘∅)) | |
3 | 1, 2 | oveq12d 7177 | . . 3 ⊢ (𝑋 = ∅ → (𝑋 + ( ⊥ ‘𝑋)) = (∅ + ( ⊥ ‘∅))) |
4 | pexmidALT.a | . . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | pexmidALT.o | . . . . . . . 8 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
6 | 4, 5 | pol0N 37049 | . . . . . . 7 ⊢ (𝐾 ∈ HL → ( ⊥ ‘∅) = 𝐴) |
7 | eqimss 4026 | . . . . . . 7 ⊢ (( ⊥ ‘∅) = 𝐴 → ( ⊥ ‘∅) ⊆ 𝐴) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝐾 ∈ HL → ( ⊥ ‘∅) ⊆ 𝐴) |
9 | pexmidALT.p | . . . . . . 7 ⊢ + = (+𝑃‘𝐾) | |
10 | 4, 9 | padd02 36952 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ ( ⊥ ‘∅) ⊆ 𝐴) → (∅ + ( ⊥ ‘∅)) = ( ⊥ ‘∅)) |
11 | 8, 10 | mpdan 685 | . . . . 5 ⊢ (𝐾 ∈ HL → (∅ + ( ⊥ ‘∅)) = ( ⊥ ‘∅)) |
12 | 11, 6 | eqtrd 2859 | . . . 4 ⊢ (𝐾 ∈ HL → (∅ + ( ⊥ ‘∅)) = 𝐴) |
13 | 12 | ad2antrr 724 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (∅ + ( ⊥ ‘∅)) = 𝐴) |
14 | 3, 13 | sylan9eqr 2881 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) ∧ 𝑋 = ∅) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴) |
15 | 4, 9, 5 | pexmidlem8N 37117 | . . 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 3107 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ⊆ wss 3939 ∅c0 4294 ‘cfv 6358 (class class class)co 7159 Atomscatm 36403 HLchlt 36490 +𝑃cpadd 36935 ⊥𝑃cpolN 37042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-riotaBAD 36093 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-undef 7942 df-proset 17541 df-poset 17559 df-plt 17571 df-lub 17587 df-glb 17588 df-join 17589 df-meet 17590 df-p0 17652 df-p1 17653 df-lat 17659 df-clat 17721 df-oposet 36316 df-ol 36318 df-oml 36319 df-covers 36406 df-ats 36407 df-atl 36438 df-cvlat 36462 df-hlat 36491 df-psubsp 36643 df-pmap 36644 df-padd 36936 df-polarityN 37043 df-psubclN 37075 |
This theorem is referenced by: (None) |
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