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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 473 | . . 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 34471 | . . . 4 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
5 | 4 | adantr 481 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
6 | simpr 477 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
7 | eqid 2622 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | opltne0.s | . . . 4 ⊢ < = (lt‘𝐾) | |
9 | 7, 8 | pltval 16960 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ ( 0 (le‘𝐾)𝑋 ∧ 0 ≠ 𝑋))) |
10 | 1, 5, 6, 9 | syl3anc 1326 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ ( 0 (le‘𝐾)𝑋 ∧ 0 ≠ 𝑋))) |
11 | necom 2847 | . . 3 ⊢ (𝑋 ≠ 0 ↔ 0 ≠ 𝑋) | |
12 | 2, 7, 3 | op0le 34473 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 (le‘𝐾)𝑋) |
13 | 12 | biantrurd 529 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( 0 ≠ 𝑋 ↔ ( 0 (le‘𝐾)𝑋 ∧ 0 ≠ 𝑋))) |
14 | 11, 13 | syl5rbb 273 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → (( 0 (le‘𝐾)𝑋 ∧ 0 ≠ 𝑋) ↔ 𝑋 ≠ 0 )) |
15 | 10, 14 | bitrd 268 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ 𝑋 ≠ 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 ‘cfv 5888 Basecbs 15857 lecple 15948 ltcplt 16941 0.cp0 17037 OPcops 34459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-plt 16958 df-glb 16975 df-p0 17039 df-oposet 34463 |
This theorem is referenced by: atle 34722 dalemcea 34946 2atm2atN 35071 dia2dimlem2 36354 dia2dimlem3 36355 |
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