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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 2726 | . . 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 38772 | . 2 ⊢ (𝐾 ∈ HL → ∃𝑥 ∈ (Base‘𝐾)( 0 < 𝑥 ∧ 𝑥 < 1 )) |
6 | hlpos 38748 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset) | |
7 | 6 | adantr 480 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐾 ∈ Poset) |
8 | hlop 38744 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
9 | 8 | adantr 480 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 𝐾 ∈ OP) |
10 | 1, 3 | op0cl 38566 | . . . . 5 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾)) |
11 | 9, 10 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 0 ∈ (Base‘𝐾)) |
12 | simpr 484 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 𝑥 ∈ (Base‘𝐾)) | |
13 | 1, 4 | op1cl 38567 | . . . . 5 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾)) |
14 | 9, 13 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑥 ∈ (Base‘𝐾)) → 1 ∈ (Base‘𝐾)) |
15 | 1, 2 | plttr 18304 | . . . 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 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 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 class class class wbr 5141 ‘cfv 6536 Basecbs 17150 Posetcpo 18269 ltcplt 18270 0.cp0 18385 1.cp1 18386 OPcops 38554 HLchlt 38732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-proset 18257 df-poset 18275 df-plt 18292 df-lub 18308 df-glb 18309 df-p0 18387 df-p1 18388 df-lat 18394 df-oposet 38558 df-ol 38560 df-oml 38561 df-atl 38680 df-cvlat 38704 df-hlat 38733 |
This theorem is referenced by: (None) |
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