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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opltn0 | Structured version Visualization version GIF version |
Description: A lattice element greater than zero is nonzero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.) |
Ref | Expression |
---|---|
opltne0.b | ⊢ 𝐵 = (Base‘𝐾) |
opltne0.s | ⊢ < = (lt‘𝐾) |
opltne0.z | ⊢ 0 = (0.‘𝐾) |
Ref | Expression |
---|---|
opltn0 | ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ 𝑋 ≠ 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 486 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OP) | |
2 | opltne0.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
3 | opltne0.z | . . . . 5 ⊢ 0 = (0.‘𝐾) | |
4 | 2, 3 | op0cl 36480 | . . . 4 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
5 | 4 | adantr 484 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
6 | simpr 488 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
7 | eqid 2798 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | opltne0.s | . . . 4 ⊢ < = (lt‘𝐾) | |
9 | 7, 8 | pltval 17562 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ ( 0 (le‘𝐾)𝑋 ∧ 0 ≠ 𝑋))) |
10 | 1, 5, 6, 9 | syl3anc 1368 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ ( 0 (le‘𝐾)𝑋 ∧ 0 ≠ 𝑋))) |
11 | necom 3040 | . . 3 ⊢ (𝑋 ≠ 0 ↔ 0 ≠ 𝑋) | |
12 | 2, 7, 3 | op0le 36482 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
13 | 12 | biantrurd 536 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( 0 ≠ 𝑋 ↔ ( 0 (le‘𝐾)𝑋 ∧ 0 ≠ 𝑋))) |
14 | 11, 13 | syl5rbb 287 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (( 0 (le‘𝐾)𝑋 ∧ 0 ≠ 𝑋) ↔ 𝑋 ≠ 0 )) |
15 | 10, 14 | bitrd 282 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ 𝑋 ≠ 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 class class class wbr 5030 ‘cfv 6324 Basecbs 16475 lecple 16564 ltcplt 17543 0.cp0 17639 OPcops 36468 |
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-plt 17560 df-glb 17577 df-p0 17641 df-oposet 36472 |
This theorem is referenced by: atle 36732 dalemcea 36956 2atm2atN 37081 dia2dimlem2 38361 dia2dimlem3 38362 |
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