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Theorem cvrexchlem 36107
Description: Lemma for cvrexch 36108. (cvexchlem 29832 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 36051 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2 cvrexch.b . . . . . . . 8 𝐵 = (Base‘𝐾)
3 cvrexch.m . . . . . . . 8 = (meet‘𝐾)
42, 3latmcl 17495 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
51, 4syl3an1 1156 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
6 eqid 2797 . . . . . . . 8 (lt‘𝐾) = (lt‘𝐾)
7 cvrexch.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
82, 6, 7cvrlt 35958 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋 𝑌) ∈ 𝐵𝑌𝐵) ∧ (𝑋 𝑌)𝐶𝑌) → (𝑋 𝑌)(lt‘𝐾)𝑌)
98ex 413 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 𝑌) ∈ 𝐵𝑌𝐵) → ((𝑋 𝑌)𝐶𝑌 → (𝑋 𝑌)(lt‘𝐾)𝑌))
105, 9syld3an2 1404 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌)𝐶𝑌 → (𝑋 𝑌)(lt‘𝐾)𝑌))
11 eqid 2797 . . . . . . 7 (le‘𝐾) = (le‘𝐾)
12 eqid 2797 . . . . . . 7 (Atoms‘𝐾) = (Atoms‘𝐾)
132, 11, 6, 12hlrelat1 36088 . . . . . 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 1404 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌)(lt‘𝐾)𝑌 → ∃𝑝 ∈ (Atoms‘𝐾)(¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌)))
1510, 14syld 47 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌)𝐶𝑌 → ∃𝑝 ∈ (Atoms‘𝐾)(¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌)))
1615imp 407 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌)𝐶𝑌) → ∃𝑝 ∈ (Atoms‘𝐾)(¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌))
17 simpl1 1184 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ HL)
1817hllatd 36052 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ Lat)
192, 12atbase 35977 . . . . . . . . . . . . . . . . 17 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
2019adantl 482 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝𝐵)
21 simpl2 1185 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑋𝐵)
22 simpl3 1186 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑌𝐵)
232, 11, 3latlem12 17521 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑋𝐵𝑌𝐵)) → ((𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)𝑌) ↔ 𝑝(le‘𝐾)(𝑋 𝑌)))
2418, 20, 21, 22, 23syl13anc 1365 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)𝑌) ↔ 𝑝(le‘𝐾)(𝑋 𝑌)))
2524biimpd 230 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)𝑌) → 𝑝(le‘𝐾)(𝑋 𝑌)))
2625expcomd 417 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝(le‘𝐾)𝑌 → (𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)(𝑋 𝑌))))
27 con3 156 . . . . . . . . . . . . 13 ((𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)(𝑋 𝑌)) → (¬ 𝑝(le‘𝐾)(𝑋 𝑌) → ¬ 𝑝(le‘𝐾)𝑋))
2826, 27syl6 35 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝(le‘𝐾)𝑌 → (¬ 𝑝(le‘𝐾)(𝑋 𝑌) → ¬ 𝑝(le‘𝐾)𝑋)))
2928com23 86 . . . . . . . . . . 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 425 . . . . . . . . 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 485 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ (Atoms‘𝐾))
33 cvrexch.j . . . . . . . . . . 11 = (join‘𝐾)
342, 11, 33, 7, 12cvr1 36098 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑝 ∈ (Atoms‘𝐾)) → (¬ 𝑝(le‘𝐾)𝑋𝑋𝐶(𝑋 𝑝)))
3517, 21, 32, 34syl3anc 1364 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (¬ 𝑝(le‘𝐾)𝑋𝑋𝐶(𝑋 𝑝)))
3631, 35sylibd 240 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (((𝑋 𝑌)𝐶𝑌 ∧ (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌)) → 𝑋𝐶(𝑋 𝑝)))
3736imp 407 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ ((𝑋 𝑌)𝐶𝑌 ∧ (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌))) → 𝑋𝐶(𝑋 𝑝))
38 simpl1 1184 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝐾 ∈ HL)
3938hllatd 36052 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝐾 ∈ Lat)
40 simpl2 1185 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑋𝐵)
41 simpl3 1186 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑌𝐵)
4239, 40, 41, 4syl3anc 1364 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑋 𝑌) ∈ 𝐵)
43 simpr 485 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑝𝐵)
442, 33latjass 17538 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑝𝐵)) → ((𝑋 (𝑋 𝑌)) 𝑝) = (𝑋 ((𝑋 𝑌) 𝑝)))
4539, 40, 42, 43, 44syl13anc 1365 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑋 (𝑋 𝑌)) 𝑝) = (𝑋 ((𝑋 𝑌) 𝑝)))
462, 33, 3latabs1 17530 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)
471, 46syl3an1 1156 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)
4847adantr 481 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)
4948oveq1d 7038 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑋 (𝑋 𝑌)) 𝑝) = (𝑋 𝑝))
5045, 49eqtr3d 2835 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑋 ((𝑋 𝑌) 𝑝)) = (𝑋 𝑝))
5150adantr 481 . . . . . . . . 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 17528 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵𝑝𝐵) → (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ↔ (𝑋 𝑌)(lt‘𝐾)((𝑋 𝑌) 𝑝)))
5339, 42, 43, 52syl3anc 1364 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ↔ (𝑋 𝑌)(lt‘𝐾)((𝑋 𝑌) 𝑝)))
542, 11, 3latmle2 17520 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑌)
5539, 40, 41, 54syl3anc 1364 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑋 𝑌)(le‘𝐾)𝑌)
5655biantrurd 533 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑝(le‘𝐾)𝑌 ↔ ((𝑋 𝑌)(le‘𝐾)𝑌𝑝(le‘𝐾)𝑌)))
572, 11, 33latjle12 17505 . . . . . . . . . . . . . . . 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 1365 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (((𝑋 𝑌)(le‘𝐾)𝑌𝑝(le‘𝐾)𝑌) ↔ ((𝑋 𝑌) 𝑝)(le‘𝐾)𝑌))
5956, 58bitrd 280 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑝(le‘𝐾)𝑌 ↔ ((𝑋 𝑌) 𝑝)(le‘𝐾)𝑌))
6053, 59anbi12d 630 . . . . . . . . . . . . 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 36054 . . . . . . . . . . . . . . . 16 (𝐾 ∈ HL → 𝐾 ∈ Poset)
6238, 61syl 17 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝐾 ∈ Poset)
632, 33latjcl 17494 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵𝑝𝐵) → ((𝑋 𝑌) 𝑝) ∈ 𝐵)
6439, 42, 43, 63syl3anc 1364 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑋 𝑌) 𝑝) ∈ 𝐵)
6542, 41, 643jca 1121 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑋 𝑌) ∈ 𝐵𝑌𝐵 ∧ ((𝑋 𝑌) 𝑝) ∈ 𝐵))
662, 11, 6, 7cvrnbtwn2 35963 . . . . . . . . . . . . . . . . 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 230 . . . . . . . . . . . . . . . 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 1112 . . . . . . . . . . . . . . 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 65 . . . . . . . . . . . . . 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 86 . . . . . . . . . . . . 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 241 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌) → ((𝑋 𝑌)𝐶𝑌 → ((𝑋 𝑌) 𝑝) = 𝑌)))
7271com23 86 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑋 𝑌)𝐶𝑌 → ((¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌) → ((𝑋 𝑌) 𝑝) = 𝑌)))
7372imp32 419 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ ((𝑋 𝑌)𝐶𝑌 ∧ (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌))) → ((𝑋 𝑌) 𝑝) = 𝑌)
7473oveq2d 7039 . . . . . . . . 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 2835 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ ((𝑋 𝑌)𝐶𝑌 ∧ (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌))) → (𝑋 𝑝) = (𝑋 𝑌))
7619, 75sylanl2 677 . . . . . . 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 4994 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ ((𝑋 𝑌)𝐶𝑌 ∧ (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌))) → 𝑋𝐶(𝑋 𝑌))
7877expr 457 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ (𝑋 𝑌)𝐶𝑌) → ((¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌) → 𝑋𝐶(𝑋 𝑌)))
7978an32s 648 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌)𝐶𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌) → 𝑋𝐶(𝑋 𝑌)))
8079rexlimdva 3249 . . 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 413 1 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌)𝐶𝑌𝑋𝐶(𝑋 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1080   = wceq 1525  wcel 2083  wrex 3108   class class class wbr 4968  cfv 6232  (class class class)co 7023  Basecbs 16316  lecple 16405  Posetcpo 17383  ltcplt 17384  joincjn 17387  meetcmee 17388  Latclat 17488  ccvr 35950  Atomscatm 35951  HLchlt 36038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-proset 17371  df-poset 17389  df-plt 17401  df-lub 17417  df-glb 17418  df-join 17419  df-meet 17420  df-p0 17482  df-lat 17489  df-clat 17551  df-oposet 35864  df-ol 35866  df-oml 35867  df-covers 35954  df-ats 35955  df-atl 35986  df-cvlat 36010  df-hlat 36039
This theorem is referenced by:  cvrexch  36108
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