# Metamath Proof Explorer

## Theorem cvlatexchb1

Description: A version of cvlexchb1 for atoms. (Contributed by NM, 5-Nov-2012)

Ref Expression
Hypotheses cvlatexch.l
cvlatexch.j
cvlatexch.a ${⊢}{A}=\mathrm{Atoms}\left({K}\right)$
Assertion cvlatexchb1

### Proof

Step Hyp Ref Expression
1 cvlatexch.l
2 cvlatexch.j
3 cvlatexch.a ${⊢}{A}=\mathrm{Atoms}\left({K}\right)$
4 cvlatl ${⊢}{K}\in \mathrm{CvLat}\to {K}\in \mathrm{AtLat}$
5 4 adantr ${⊢}\left({K}\in \mathrm{CvLat}\wedge \left({P}\in {A}\wedge {Q}\in {A}\wedge {R}\in {A}\right)\right)\to {K}\in \mathrm{AtLat}$
6 simpr1 ${⊢}\left({K}\in \mathrm{CvLat}\wedge \left({P}\in {A}\wedge {Q}\in {A}\wedge {R}\in {A}\right)\right)\to {P}\in {A}$
7 simpr3 ${⊢}\left({K}\in \mathrm{CvLat}\wedge \left({P}\in {A}\wedge {Q}\in {A}\wedge {R}\in {A}\right)\right)\to {R}\in {A}$
8 1 3 atncmp
9 5 6 7 8 syl3anc
10 eqid ${⊢}{\mathrm{Base}}_{{K}}={\mathrm{Base}}_{{K}}$
11 10 3 atbase ${⊢}{R}\in {A}\to {R}\in {\mathrm{Base}}_{{K}}$
12 10 1 2 3 cvlexchb1
13 12 3expia
14 11 13 syl3anr3
15 9 14 sylbird
16 15 3impia