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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 2798 | . . 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 36685 | . 2 ⊢ (𝐾 ∈ HL → ∃𝑥 ∈ (Base‘𝐾)( 0 < 𝑥 ∧ 𝑥 < 1 )) |
6 | hlpos 36662 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset) | |
7 | 6 | adantr 484 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐾 ∈ Poset) |
8 | hlop 36658 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
9 | 8 | adantr 484 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐾 ∈ OP) |
10 | 1, 3 | op0cl 36480 | . . . . 5 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾)) |
11 | 9, 10 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 0 ∈ (Base‘𝐾)) |
12 | simpr 488 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 𝑥 ∈ (Base‘𝐾)) | |
13 | 1, 4 | op1cl 36481 | . . . . 5 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾)) |
14 | 9, 13 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 1 ∈ (Base‘𝐾)) |
15 | 1, 2 | plttr 17572 | . . . 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 1369 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → (( 0 < 𝑥 ∧ 𝑥 < 1 ) → 0 < 1 )) |
17 | 16 | rexlimdva 3243 | . 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 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 class class class wbr 5030 ‘cfv 6324 Basecbs 16475 Posetcpo 17542 ltcplt 17543 0.cp0 17639 1.cp1 17640 OPcops 36468 HLchlt 36646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-p0 17641 df-p1 17642 df-lat 17648 df-oposet 36472 df-ol 36474 df-oml 36475 df-atl 36594 df-cvlat 36618 df-hlat 36647 |
This theorem is referenced by: (None) |
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