0ltat - Mathbox for Norm Megill

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Theorem 0ltat 36442

Description: An atom is greater than zero. (Contributed by NM, 4-Jul-2012.)

**Hypotheses**
Ref	Expression
0ltat.z	⊢ 0 = (0.‘𝐾)
0ltat.s	⊢ < = (lt‘𝐾)
0ltat.a	⊢ 𝐴 = (Atoms‘𝐾)

**Assertion**
Ref	Expression
0ltat	⊢ ((𝐾 ∈ OP ∧ 𝑃 ∈ 𝐴) → 0 < 𝑃)

**Proof of Theorem 0ltat**
Step	Hyp	Ref	Expression
1		simpl 485	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑃 ∈ 𝐴) → 𝐾 ∈ OP)
2		eqid 2821	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		0ltat.z	. . . 4 ⊢ 0 = (0.‘𝐾)
4	2, 3	op0cl 36335	. . 3 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾))
5	4	adantr 483	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑃 ∈ 𝐴) → 0 ∈ (Base‘𝐾))
6		0ltat.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
7	2, 6	atbase 36440	. . 3 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
8	7	adantl 484	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ (Base‘𝐾))
9		eqid 2821	. . 3 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
10	3, 9, 6	atcvr0 36439	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑃 ∈ 𝐴) → 0 ( ⋖ ‘𝐾)𝑃)
11		0ltat.s	. . 3 ⊢ < = (lt‘𝐾)
12	2, 11, 9	cvrlt 36421	. 2 ⊢ (((𝐾 ∈ OP ∧ 0 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) ∧ 0 ( ⋖ ‘𝐾)𝑃) → 0 < 𝑃)
13	1, 5, 8, 10, 12	syl31anc 1369	1 ⊢ ((𝐾 ∈ OP ∧ 𝑃 ∈ 𝐴) → 0 < 𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 Basecbs 16483 ltcplt 17551 0.cp0 17647 OPcops 36323 ⋖ ccvr 36413 Atomscatm 36414

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-glb 17585 df-p0 17649 df-oposet 36327 df-covers 36417 df-ats 36418

This theorem is referenced by: 2atm2atN 36936 dia2dimlem2 38216 dia2dimlem3 38217

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