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Theorem cvrexchlem 33526
Description: Lemma for cvrexch 33527. (cvexchlem 28417 analog.) (Contributed by NM, 18-Nov-2011.)
Hypotheses
Ref Expression
cvrexch.b 𝐵 = (Base‘𝐾)
cvrexch.j = (join‘𝐾)
cvrexch.m = (meet‘𝐾)
cvrexch.c 𝐶 = ( ⋖ ‘𝐾)
Assertion
Ref Expression
cvrexchlem ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌)𝐶𝑌𝑋𝐶(𝑋 𝑌)))

Proof of Theorem cvrexchlem
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 hllat 33471 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2 cvrexch.b . . . . . . . 8 𝐵 = (Base‘𝐾)
3 cvrexch.m . . . . . . . 8 = (meet‘𝐾)
42, 3latmcl 16821 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
51, 4syl3an1 1350 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
6 eqid 2609 . . . . . . . 8 (lt‘𝐾) = (lt‘𝐾)
7 cvrexch.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
82, 6, 7cvrlt 33378 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋 𝑌) ∈ 𝐵𝑌𝐵) ∧ (𝑋 𝑌)𝐶𝑌) → (𝑋 𝑌)(lt‘𝐾)𝑌)
98ex 448 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 𝑌) ∈ 𝐵𝑌𝐵) → ((𝑋 𝑌)𝐶𝑌 → (𝑋 𝑌)(lt‘𝐾)𝑌))
105, 9syld3an2 1364 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌)𝐶𝑌 → (𝑋 𝑌)(lt‘𝐾)𝑌))
11 eqid 2609 . . . . . . 7 (le‘𝐾) = (le‘𝐾)
12 eqid 2609 . . . . . . 7 (Atoms‘𝐾) = (Atoms‘𝐾)
132, 11, 6, 12hlrelat1 33507 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 𝑌) ∈ 𝐵𝑌𝐵) → ((𝑋 𝑌)(lt‘𝐾)𝑌 → ∃𝑝 ∈ (Atoms‘𝐾)(¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌)))
145, 13syld3an2 1364 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌)(lt‘𝐾)𝑌 → ∃𝑝 ∈ (Atoms‘𝐾)(¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌)))
1510, 14syld 45 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌)𝐶𝑌 → ∃𝑝 ∈ (Atoms‘𝐾)(¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌)))
1615imp 443 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌)𝐶𝑌) → ∃𝑝 ∈ (Atoms‘𝐾)(¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌))
17 simpl1 1056 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ HL)
1817, 1syl 17 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ Lat)
192, 12atbase 33397 . . . . . . . . . . . . . . . . 17 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
2019adantl 480 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝𝐵)
21 simpl2 1057 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑋𝐵)
22 simpl3 1058 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑌𝐵)
232, 11, 3latlem12 16847 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑋𝐵𝑌𝐵)) → ((𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)𝑌) ↔ 𝑝(le‘𝐾)(𝑋 𝑌)))
2418, 20, 21, 22, 23syl13anc 1319 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)𝑌) ↔ 𝑝(le‘𝐾)(𝑋 𝑌)))
2524biimpd 217 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)𝑌) → 𝑝(le‘𝐾)(𝑋 𝑌)))
2625expcomd 452 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝(le‘𝐾)𝑌 → (𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)(𝑋 𝑌))))
27 con3 147 . . . . . . . . . . . . 13 ((𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)(𝑋 𝑌)) → (¬ 𝑝(le‘𝐾)(𝑋 𝑌) → ¬ 𝑝(le‘𝐾)𝑋))
2826, 27syl6 34 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝(le‘𝐾)𝑌 → (¬ 𝑝(le‘𝐾)(𝑋 𝑌) → ¬ 𝑝(le‘𝐾)𝑋)))
2928com23 83 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (¬ 𝑝(le‘𝐾)(𝑋 𝑌) → (𝑝(le‘𝐾)𝑌 → ¬ 𝑝(le‘𝐾)𝑋)))
3029a1d 25 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((𝑋 𝑌)𝐶𝑌 → (¬ 𝑝(le‘𝐾)(𝑋 𝑌) → (𝑝(le‘𝐾)𝑌 → ¬ 𝑝(le‘𝐾)𝑋))))
3130imp4d 615 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (((𝑋 𝑌)𝐶𝑌 ∧ (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌)) → ¬ 𝑝(le‘𝐾)𝑋))
32 simpr 475 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ (Atoms‘𝐾))
33 cvrexch.j . . . . . . . . . . 11 = (join‘𝐾)
342, 11, 33, 7, 12cvr1 33517 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑝 ∈ (Atoms‘𝐾)) → (¬ 𝑝(le‘𝐾)𝑋𝑋𝐶(𝑋 𝑝)))
3517, 21, 32, 34syl3anc 1317 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (¬ 𝑝(le‘𝐾)𝑋𝑋𝐶(𝑋 𝑝)))
3631, 35sylibd 227 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (((𝑋 𝑌)𝐶𝑌 ∧ (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌)) → 𝑋𝐶(𝑋 𝑝)))
3736imp 443 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ ((𝑋 𝑌)𝐶𝑌 ∧ (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌))) → 𝑋𝐶(𝑋 𝑝))
38 simpl1 1056 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝐾 ∈ HL)
3938, 1syl 17 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝐾 ∈ Lat)
40 simpl2 1057 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑋𝐵)
41 simpl3 1058 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑌𝐵)
4239, 40, 41, 4syl3anc 1317 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑋 𝑌) ∈ 𝐵)
43 simpr 475 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑝𝐵)
442, 33latjass 16864 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑝𝐵)) → ((𝑋 (𝑋 𝑌)) 𝑝) = (𝑋 ((𝑋 𝑌) 𝑝)))
4539, 40, 42, 43, 44syl13anc 1319 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑋 (𝑋 𝑌)) 𝑝) = (𝑋 ((𝑋 𝑌) 𝑝)))
462, 33, 3latabs1 16856 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)
471, 46syl3an1 1350 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)
4847adantr 479 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)
4948oveq1d 6542 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑋 (𝑋 𝑌)) 𝑝) = (𝑋 𝑝))
5045, 49eqtr3d 2645 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑋 ((𝑋 𝑌) 𝑝)) = (𝑋 𝑝))
5150adantr 479 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ ((𝑋 𝑌)𝐶𝑌 ∧ (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌))) → (𝑋 ((𝑋 𝑌) 𝑝)) = (𝑋 𝑝))
522, 11, 6, 33latnle 16854 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵𝑝𝐵) → (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ↔ (𝑋 𝑌)(lt‘𝐾)((𝑋 𝑌) 𝑝)))
5339, 42, 43, 52syl3anc 1317 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ↔ (𝑋 𝑌)(lt‘𝐾)((𝑋 𝑌) 𝑝)))
542, 11, 3latmle2 16846 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑌)
5539, 40, 41, 54syl3anc 1317 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑋 𝑌)(le‘𝐾)𝑌)
5655biantrurd 527 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑝(le‘𝐾)𝑌 ↔ ((𝑋 𝑌)(le‘𝐾)𝑌𝑝(le‘𝐾)𝑌)))
572, 11, 33latjle12 16831 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ ((𝑋 𝑌) ∈ 𝐵𝑝𝐵𝑌𝐵)) → (((𝑋 𝑌)(le‘𝐾)𝑌𝑝(le‘𝐾)𝑌) ↔ ((𝑋 𝑌) 𝑝)(le‘𝐾)𝑌))
5839, 42, 43, 41, 57syl13anc 1319 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (((𝑋 𝑌)(le‘𝐾)𝑌𝑝(le‘𝐾)𝑌) ↔ ((𝑋 𝑌) 𝑝)(le‘𝐾)𝑌))
5956, 58bitrd 266 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑝(le‘𝐾)𝑌 ↔ ((𝑋 𝑌) 𝑝)(le‘𝐾)𝑌))
6053, 59anbi12d 742 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌) ↔ ((𝑋 𝑌)(lt‘𝐾)((𝑋 𝑌) 𝑝) ∧ ((𝑋 𝑌) 𝑝)(le‘𝐾)𝑌)))
61 hlpos 33473 . . . . . . . . . . . . . . . 16 (𝐾 ∈ HL → 𝐾 ∈ Poset)
6238, 61syl 17 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝐾 ∈ Poset)
632, 33latjcl 16820 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵𝑝𝐵) → ((𝑋 𝑌) 𝑝) ∈ 𝐵)
6439, 42, 43, 63syl3anc 1317 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑋 𝑌) 𝑝) ∈ 𝐵)
6542, 41, 643jca 1234 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑋 𝑌) ∈ 𝐵𝑌𝐵 ∧ ((𝑋 𝑌) 𝑝) ∈ 𝐵))
662, 11, 6, 7cvrnbtwn2 33383 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Poset ∧ ((𝑋 𝑌) ∈ 𝐵𝑌𝐵 ∧ ((𝑋 𝑌) 𝑝) ∈ 𝐵) ∧ (𝑋 𝑌)𝐶𝑌) → (((𝑋 𝑌)(lt‘𝐾)((𝑋 𝑌) 𝑝) ∧ ((𝑋 𝑌) 𝑝)(le‘𝐾)𝑌) ↔ ((𝑋 𝑌) 𝑝) = 𝑌))
6766biimpd 217 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Poset ∧ ((𝑋 𝑌) ∈ 𝐵𝑌𝐵 ∧ ((𝑋 𝑌) 𝑝) ∈ 𝐵) ∧ (𝑋 𝑌)𝐶𝑌) → (((𝑋 𝑌)(lt‘𝐾)((𝑋 𝑌) 𝑝) ∧ ((𝑋 𝑌) 𝑝)(le‘𝐾)𝑌) → ((𝑋 𝑌) 𝑝) = 𝑌))
68673exp 1255 . . . . . . . . . . . . . . 15 (𝐾 ∈ Poset → (((𝑋 𝑌) ∈ 𝐵𝑌𝐵 ∧ ((𝑋 𝑌) 𝑝) ∈ 𝐵) → ((𝑋 𝑌)𝐶𝑌 → (((𝑋 𝑌)(lt‘𝐾)((𝑋 𝑌) 𝑝) ∧ ((𝑋 𝑌) 𝑝)(le‘𝐾)𝑌) → ((𝑋 𝑌) 𝑝) = 𝑌))))
6962, 65, 68sylc 62 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑋 𝑌)𝐶𝑌 → (((𝑋 𝑌)(lt‘𝐾)((𝑋 𝑌) 𝑝) ∧ ((𝑋 𝑌) 𝑝)(le‘𝐾)𝑌) → ((𝑋 𝑌) 𝑝) = 𝑌)))
7069com23 83 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (((𝑋 𝑌)(lt‘𝐾)((𝑋 𝑌) 𝑝) ∧ ((𝑋 𝑌) 𝑝)(le‘𝐾)𝑌) → ((𝑋 𝑌)𝐶𝑌 → ((𝑋 𝑌) 𝑝) = 𝑌)))
7160, 70sylbid 228 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌) → ((𝑋 𝑌)𝐶𝑌 → ((𝑋 𝑌) 𝑝) = 𝑌)))
7271com23 83 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑋 𝑌)𝐶𝑌 → ((¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌) → ((𝑋 𝑌) 𝑝) = 𝑌)))
7372imp32 447 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ ((𝑋 𝑌)𝐶𝑌 ∧ (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌))) → ((𝑋 𝑌) 𝑝) = 𝑌)
7473oveq2d 6543 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ ((𝑋 𝑌)𝐶𝑌 ∧ (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌))) → (𝑋 ((𝑋 𝑌) 𝑝)) = (𝑋 𝑌))
7551, 74eqtr3d 2645 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ ((𝑋 𝑌)𝐶𝑌 ∧ (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌))) → (𝑋 𝑝) = (𝑋 𝑌))
7619, 75sylanl2 680 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ ((𝑋 𝑌)𝐶𝑌 ∧ (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌))) → (𝑋 𝑝) = (𝑋 𝑌))
7737, 76breqtrd 4603 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ ((𝑋 𝑌)𝐶𝑌 ∧ (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌))) → 𝑋𝐶(𝑋 𝑌))
7877expr 640 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ (𝑋 𝑌)𝐶𝑌) → ((¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌) → 𝑋𝐶(𝑋 𝑌)))
7978an32s 841 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌)𝐶𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌) → 𝑋𝐶(𝑋 𝑌)))
8079rexlimdva 3012 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌)𝐶𝑌) → (∃𝑝 ∈ (Atoms‘𝐾)(¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌) → 𝑋𝐶(𝑋 𝑌)))
8116, 80mpd 15 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌)𝐶𝑌) → 𝑋𝐶(𝑋 𝑌))
8281ex 448 1 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌)𝐶𝑌𝑋𝐶(𝑋 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wrex 2896   class class class wbr 4577  cfv 5790  (class class class)co 6527  Basecbs 15641  lecple 15721  Posetcpo 16709  ltcplt 16710  joincjn 16713  meetcmee 16714  Latclat 16814  ccvr 33370  Atomscatm 33371  HLchlt 33458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-preset 16697  df-poset 16715  df-plt 16727  df-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-p0 16808  df-lat 16815  df-clat 16877  df-oposet 33284  df-ol 33286  df-oml 33287  df-covers 33374  df-ats 33375  df-atl 33406  df-cvlat 33430  df-hlat 33459
This theorem is referenced by:  cvrexch  33527
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