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Mirrors > Home > MPE Home > Th. List > Mathboxes > hl0lt1N | Structured version Visualization version GIF version |
Description: Lattice 0 is less than lattice 1 in a Hilbert lattice. (Contributed by NM, 4-Dec-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hl0lt1.s | ⊢ < = (lt‘𝐾) |
hl0lt1.z | ⊢ 0 = (0.‘𝐾) |
hl0lt1.u | ⊢ 1 = (1.‘𝐾) |
Ref | Expression |
---|---|
hl0lt1N | ⊢ (𝐾 ∈ HL → 0 < 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | hl0lt1.s | . . 3 ⊢ < = (lt‘𝐾) | |
3 | hl0lt1.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
4 | hl0lt1.u | . . 3 ⊢ 1 = (1.‘𝐾) | |
5 | 1, 2, 3, 4 | hlhgt2 36519 | . 2 ⊢ (𝐾 ∈ HL → ∃𝑥 ∈ (Base‘𝐾)( 0 < 𝑥 ∧ 𝑥 < 1 )) |
6 | hlpos 36496 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset) | |
7 | 6 | adantr 483 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐾 ∈ Poset) |
8 | hlop 36492 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
9 | 8 | adantr 483 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐾 ∈ OP) |
10 | 1, 3 | op0cl 36314 | . . . . 5 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾)) |
11 | 9, 10 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 0 ∈ (Base‘𝐾)) |
12 | simpr 487 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 𝑥 ∈ (Base‘𝐾)) | |
13 | 1, 4 | op1cl 36315 | . . . . 5 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾)) |
14 | 9, 13 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 1 ∈ (Base‘𝐾)) |
15 | 1, 2 | plttr 17574 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ ( 0 ∈ (Base‘𝐾) ∧ 𝑥 ∈ (Base‘𝐾) ∧ 1 ∈ (Base‘𝐾))) → (( 0 < 𝑥 ∧ 𝑥 < 1 ) → 0 < 1 )) |
16 | 7, 11, 12, 14, 15 | syl13anc 1368 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → (( 0 < 𝑥 ∧ 𝑥 < 1 ) → 0 < 1 )) |
17 | 16 | rexlimdva 3284 | . 2 ⊢ (𝐾 ∈ HL → (∃𝑥 ∈ (Base‘𝐾)( 0 < 𝑥 ∧ 𝑥 < 1 ) → 0 < 1 )) |
18 | 5, 17 | mpd 15 | 1 ⊢ (𝐾 ∈ HL → 0 < 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 class class class wbr 5058 ‘cfv 6349 Basecbs 16477 Posetcpo 17544 ltcplt 17545 0.cp0 17641 1.cp1 17642 OPcops 36302 HLchlt 36480 |
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 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-p0 17643 df-p1 17644 df-lat 17650 df-oposet 36306 df-ol 36308 df-oml 36309 df-atl 36428 df-cvlat 36452 df-hlat 36481 |
This theorem is referenced by: (None) |
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