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Theorem cvrexchlem 39421
Description: Lemma for cvrexch 39422. (cvexchlem 32387 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 39364 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ Lat)
2 cvrexch.b . . . . . . . 8 𝐵 = (Base‘𝐾)
3 cvrexch.m . . . . . . . 8 = (meet‘𝐾)
42, 3latmcl 18485 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
51, 4syl3an1 1164 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
6 eqid 2737 . . . . . . . 8 (lt‘𝐾) = (lt‘𝐾)
7 cvrexch.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
82, 6, 7cvrlt 39271 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋 𝑌) ∈ 𝐵𝑌𝐵) ∧ (𝑋 𝑌)𝐶𝑌) → (𝑋 𝑌)(lt‘𝐾)𝑌)
98ex 412 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 𝑌) ∈ 𝐵𝑌𝐵) → ((𝑋 𝑌)𝐶𝑌 → (𝑋 𝑌)(lt‘𝐾)𝑌))
105, 9syld3an2 1413 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌)𝐶𝑌 → (𝑋 𝑌)(lt‘𝐾)𝑌))
11 eqid 2737 . . . . . . 7 (le‘𝐾) = (le‘𝐾)
12 eqid 2737 . . . . . . 7 (Atoms‘𝐾) = (Atoms‘𝐾)
132, 11, 6, 12hlrelat1 39402 . . . . . 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 1413 . . . . 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 406 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌)𝐶𝑌) → ∃𝑝 ∈ (Atoms‘𝐾)(¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌))
17 simpl1 1192 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ HL)
1817hllatd 39365 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ Lat)
192, 12atbase 39290 . . . . . . . . . . . . . . . . 17 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
2019adantl 481 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝𝐵)
21 simpl2 1193 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑋𝐵)
22 simpl3 1194 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑌𝐵)
232, 11, 3latlem12 18511 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑋𝐵𝑌𝐵)) → ((𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)𝑌) ↔ 𝑝(le‘𝐾)(𝑋 𝑌)))
2418, 20, 21, 22, 23syl13anc 1374 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)𝑌) ↔ 𝑝(le‘𝐾)(𝑋 𝑌)))
2524biimpd 229 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)𝑌) → 𝑝(le‘𝐾)(𝑋 𝑌)))
2625expcomd 416 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝(le‘𝐾)𝑌 → (𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)(𝑋 𝑌))))
27 con3 153 . . . . . . . . . . . . 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 424 . . . . . . . . 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 484 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ (Atoms‘𝐾))
33 cvrexch.j . . . . . . . . . . 11 = (join‘𝐾)
342, 11, 33, 7, 12cvr1 39412 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑝 ∈ (Atoms‘𝐾)) → (¬ 𝑝(le‘𝐾)𝑋𝑋𝐶(𝑋 𝑝)))
3517, 21, 32, 34syl3anc 1373 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (¬ 𝑝(le‘𝐾)𝑋𝑋𝐶(𝑋 𝑝)))
3631, 35sylibd 239 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (((𝑋 𝑌)𝐶𝑌 ∧ (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌)) → 𝑋𝐶(𝑋 𝑝)))
3736imp 406 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ ((𝑋 𝑌)𝐶𝑌 ∧ (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌))) → 𝑋𝐶(𝑋 𝑝))
38 simpl1 1192 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝐾 ∈ HL)
3938hllatd 39365 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝐾 ∈ Lat)
40 simpl2 1193 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑋𝐵)
41 simpl3 1194 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑌𝐵)
4239, 40, 41, 4syl3anc 1373 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑋 𝑌) ∈ 𝐵)
43 simpr 484 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑝𝐵)
442, 33latjass 18528 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ (𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵𝑝𝐵)) → ((𝑋 (𝑋 𝑌)) 𝑝) = (𝑋 ((𝑋 𝑌) 𝑝)))
4539, 40, 42, 43, 44syl13anc 1374 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑋 (𝑋 𝑌)) 𝑝) = (𝑋 ((𝑋 𝑌) 𝑝)))
462, 33, 3latabs1 18520 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)
471, 46syl3an1 1164 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)
4847adantr 480 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑋 (𝑋 𝑌)) = 𝑋)
4948oveq1d 7446 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑋 (𝑋 𝑌)) 𝑝) = (𝑋 𝑝))
5045, 49eqtr3d 2779 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑋 ((𝑋 𝑌) 𝑝)) = (𝑋 𝑝))
5150adantr 480 . . . . . . . . 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 18518 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵𝑝𝐵) → (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ↔ (𝑋 𝑌)(lt‘𝐾)((𝑋 𝑌) 𝑝)))
5339, 42, 43, 52syl3anc 1373 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ↔ (𝑋 𝑌)(lt‘𝐾)((𝑋 𝑌) 𝑝)))
542, 11, 3latmle2 18510 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌)(le‘𝐾)𝑌)
5539, 40, 41, 54syl3anc 1373 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑋 𝑌)(le‘𝐾)𝑌)
5655biantrurd 532 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑝(le‘𝐾)𝑌 ↔ ((𝑋 𝑌)(le‘𝐾)𝑌𝑝(le‘𝐾)𝑌)))
572, 11, 33latjle12 18495 . . . . . . . . . . . . . . . 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 1374 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (((𝑋 𝑌)(le‘𝐾)𝑌𝑝(le‘𝐾)𝑌) ↔ ((𝑋 𝑌) 𝑝)(le‘𝐾)𝑌))
5956, 58bitrd 279 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑝(le‘𝐾)𝑌 ↔ ((𝑋 𝑌) 𝑝)(le‘𝐾)𝑌))
6053, 59anbi12d 632 . . . . . . . . . . . . 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 39367 . . . . . . . . . . . . . . . 16 (𝐾 ∈ HL → 𝐾 ∈ Poset)
6238, 61syl 17 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝐾 ∈ Poset)
632, 33latjcl 18484 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵𝑝𝐵) → ((𝑋 𝑌) 𝑝) ∈ 𝐵)
6439, 42, 43, 63syl3anc 1373 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑋 𝑌) 𝑝) ∈ 𝐵)
6542, 41, 643jca 1129 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑋 𝑌) ∈ 𝐵𝑌𝐵 ∧ ((𝑋 𝑌) 𝑝) ∈ 𝐵))
662, 11, 6, 7cvrnbtwn2 39276 . . . . . . . . . . . . . . . . 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 229 . . . . . . . . . . . . . . . 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 1120 . . . . . . . . . . . . . . 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 240 . . . . . . . . . . . 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 418 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ ((𝑋 𝑌)𝐶𝑌 ∧ (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌))) → ((𝑋 𝑌) 𝑝) = 𝑌)
7473oveq2d 7447 . . . . . . . . 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 2779 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ ((𝑋 𝑌)𝐶𝑌 ∧ (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌))) → (𝑋 𝑝) = (𝑋 𝑌))
7619, 75sylanl2 681 . . . . . . 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 5169 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ ((𝑋 𝑌)𝐶𝑌 ∧ (¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌))) → 𝑋𝐶(𝑋 𝑌))
7877expr 456 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) ∧ (𝑋 𝑌)𝐶𝑌) → ((¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌) → 𝑋𝐶(𝑋 𝑌)))
7978an32s 652 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌)𝐶𝑌) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((¬ 𝑝(le‘𝐾)(𝑋 𝑌) ∧ 𝑝(le‘𝐾)𝑌) → 𝑋𝐶(𝑋 𝑌)))
8079rexlimdva 3155 . . 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 412 1 ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 𝑌)𝐶𝑌𝑋𝐶(𝑋 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wrex 3070   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  Posetcpo 18353  ltcplt 18354  joincjn 18357  meetcmee 18358  Latclat 18476  ccvr 39263  Atomscatm 39264  HLchlt 39351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352
This theorem is referenced by:  cvrexch  39422
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