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Mathbox for Norm Megill |
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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 2733 | . . 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 37898 | . 2 ⊢ (𝐾 ∈ HL → ∃𝑥 ∈ (Base‘𝐾)( 0 < 𝑥 ∧ 𝑥 < 1 )) |
6 | hlpos 37874 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset) | |
7 | 6 | adantr 482 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐾 ∈ Poset) |
8 | hlop 37870 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
9 | 8 | adantr 482 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐾 ∈ OP) |
10 | 1, 3 | op0cl 37692 | . . . . 5 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾)) |
11 | 9, 10 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 0 ∈ (Base‘𝐾)) |
12 | simpr 486 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 𝑥 ∈ (Base‘𝐾)) | |
13 | 1, 4 | op1cl 37693 | . . . . 5 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾)) |
14 | 9, 13 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 1 ∈ (Base‘𝐾)) |
15 | 1, 2 | plttr 18236 | . . . 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 1373 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → (( 0 < 𝑥 ∧ 𝑥 < 1 ) → 0 < 1 )) |
17 | 16 | rexlimdva 3149 | . 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 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3070 class class class wbr 5106 ‘cfv 6497 Basecbs 17088 Posetcpo 18201 ltcplt 18202 0.cp0 18317 1.cp1 18318 OPcops 37680 HLchlt 37858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-proset 18189 df-poset 18207 df-plt 18224 df-lub 18240 df-glb 18241 df-p0 18319 df-p1 18320 df-lat 18326 df-oposet 37684 df-ol 37686 df-oml 37687 df-atl 37806 df-cvlat 37830 df-hlat 37859 |
This theorem is referenced by: (None) |
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