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Theorem pmapjoin 35456
Description: The projective map of the join of two lattice elements. Part of Equation 15.5.3 of [MaedaMaeda] p. 63. (Contributed by NM, 27-Jan-2012.)
Hypotheses
Ref Expression
pmapjoin.b 𝐵 = (Base‘𝐾)
pmapjoin.j = (join‘𝐾)
pmapjoin.m 𝑀 = (pmap‘𝐾)
pmapjoin.p + = (+𝑃𝐾)
Assertion
Ref Expression
pmapjoin ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑀𝑋) + (𝑀𝑌)) ⊆ (𝑀‘(𝑋 𝑌)))

Proof of Theorem pmapjoin
Dummy variables 𝑞 𝑝 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 472 . . . . . . 7 ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) → 𝑝 ∈ (Atoms‘𝐾))
21a1i 11 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) → 𝑝 ∈ (Atoms‘𝐾)))
3 pmapjoin.b . . . . . . . 8 𝐵 = (Base‘𝐾)
4 eqid 2651 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
53, 4atbase 34894 . . . . . . 7 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
6 eqid 2651 . . . . . . . . . . 11 (le‘𝐾) = (le‘𝐾)
7 pmapjoin.j . . . . . . . . . . 11 = (join‘𝐾)
83, 6, 7latlej1 17107 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
98adantr 480 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
10 simpl1 1084 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝐾 ∈ Lat)
11 simpr 476 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑝𝐵)
12 simpl2 1085 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑋𝐵)
133, 7latjcl 17098 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
1413adantr 480 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑋 𝑌) ∈ 𝐵)
153, 6lattr 17103 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑋𝐵 ∧ (𝑋 𝑌) ∈ 𝐵)) → ((𝑝(le‘𝐾)𝑋𝑋(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
1610, 11, 12, 14, 15syl13anc 1368 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑝(le‘𝐾)𝑋𝑋(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
179, 16mpan2d 710 . . . . . . . 8 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)(𝑋 𝑌)))
1817expimpd 628 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝𝐵𝑝(le‘𝐾)𝑋) → 𝑝(le‘𝐾)(𝑋 𝑌)))
195, 18sylani 687 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) → 𝑝(le‘𝐾)(𝑋 𝑌)))
202, 19jcad 554 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) → (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)(𝑋 𝑌))))
21 simpl 472 . . . . . . 7 ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌) → 𝑝 ∈ (Atoms‘𝐾))
2221a1i 11 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌) → 𝑝 ∈ (Atoms‘𝐾)))
233, 6, 7latlej2 17108 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑌(le‘𝐾)(𝑋 𝑌))
2423adantr 480 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑌(le‘𝐾)(𝑋 𝑌))
25 simpl3 1086 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑌𝐵)
263, 6lattr 17103 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑝𝐵𝑌𝐵 ∧ (𝑋 𝑌) ∈ 𝐵)) → ((𝑝(le‘𝐾)𝑌𝑌(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
2710, 11, 25, 14, 26syl13anc 1368 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑝(le‘𝐾)𝑌𝑌(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
2824, 27mpan2d 710 . . . . . . . 8 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑝(le‘𝐾)𝑌𝑝(le‘𝐾)(𝑋 𝑌)))
2928expimpd 628 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝𝐵𝑝(le‘𝐾)𝑌) → 𝑝(le‘𝐾)(𝑋 𝑌)))
305, 29sylani 687 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌) → 𝑝(le‘𝐾)(𝑋 𝑌)))
3122, 30jcad 554 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌) → (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)(𝑋 𝑌))))
3220, 31jaod 394 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌)) → (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)(𝑋 𝑌))))
33 simpl 472 . . . . . 6 ((𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑀𝑋)∃𝑟 ∈ (𝑀𝑌)𝑝(le‘𝐾)(𝑞 𝑟)) → 𝑝 ∈ (Atoms‘𝐾))
3433a1i 11 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑀𝑋)∃𝑟 ∈ (𝑀𝑌)𝑝(le‘𝐾)(𝑞 𝑟)) → 𝑝 ∈ (Atoms‘𝐾)))
35 pmapjoin.m . . . . . . . . . . . . . 14 𝑀 = (pmap‘𝐾)
363, 6, 4, 35elpmap 35362 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑞 ∈ (𝑀𝑋) ↔ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋)))
37363adant3 1101 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑞 ∈ (𝑀𝑋) ↔ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋)))
383, 6, 4, 35elpmap 35362 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑌𝐵) → (𝑟 ∈ (𝑀𝑌) ↔ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑌)))
39383adant2 1100 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑟 ∈ (𝑀𝑌) ↔ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑌)))
4037, 39anbi12d 747 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑞 ∈ (𝑀𝑋) ∧ 𝑟 ∈ (𝑀𝑌)) ↔ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑌))))
41 an4 882 . . . . . . . . . . 11 (((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑌)) ↔ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞(le‘𝐾)𝑋𝑟(le‘𝐾)𝑌)))
4240, 41syl6bb 276 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑞 ∈ (𝑀𝑋) ∧ 𝑟 ∈ (𝑀𝑌)) ↔ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞(le‘𝐾)𝑋𝑟(le‘𝐾)𝑌))))
4342adantr 480 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑞 ∈ (𝑀𝑋) ∧ 𝑟 ∈ (𝑀𝑌)) ↔ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞(le‘𝐾)𝑋𝑟(le‘𝐾)𝑌))))
443, 4atbase 34894 . . . . . . . . . . 11 (𝑞 ∈ (Atoms‘𝐾) → 𝑞𝐵)
453, 4atbase 34894 . . . . . . . . . . 11 (𝑟 ∈ (Atoms‘𝐾) → 𝑟𝐵)
4644, 45anim12i 589 . . . . . . . . . 10 ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑞𝐵𝑟𝐵))
47 simpll1 1120 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝐾 ∈ Lat)
48 simprl 809 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑞𝐵)
49 simpll2 1121 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑋𝐵)
50 simprr 811 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑟𝐵)
51 simpll3 1122 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑌𝐵)
523, 6, 7latjlej12 17114 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ (𝑞𝐵𝑋𝐵) ∧ (𝑟𝐵𝑌𝐵)) → ((𝑞(le‘𝐾)𝑋𝑟(le‘𝐾)𝑌) → (𝑞 𝑟)(le‘𝐾)(𝑋 𝑌)))
5347, 48, 49, 50, 51, 52syl122anc 1375 . . . . . . . . . . . 12 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → ((𝑞(le‘𝐾)𝑋𝑟(le‘𝐾)𝑌) → (𝑞 𝑟)(le‘𝐾)(𝑋 𝑌)))
54 simplr 807 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑝𝐵)
553, 7latjcl 17098 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ 𝑞𝐵𝑟𝐵) → (𝑞 𝑟) ∈ 𝐵)
5647, 48, 50, 55syl3anc 1366 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → (𝑞 𝑟) ∈ 𝐵)
5713ad2antrr 762 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → (𝑋 𝑌) ∈ 𝐵)
583, 6lattr 17103 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ (𝑝𝐵 ∧ (𝑞 𝑟) ∈ 𝐵 ∧ (𝑋 𝑌) ∈ 𝐵)) → ((𝑝(le‘𝐾)(𝑞 𝑟) ∧ (𝑞 𝑟)(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
5947, 54, 56, 57, 58syl13anc 1368 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → ((𝑝(le‘𝐾)(𝑞 𝑟) ∧ (𝑞 𝑟)(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
6059expcomd 453 . . . . . . . . . . . 12 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → ((𝑞 𝑟)(le‘𝐾)(𝑋 𝑌) → (𝑝(le‘𝐾)(𝑞 𝑟) → 𝑝(le‘𝐾)(𝑋 𝑌))))
6153, 60syld 47 . . . . . . . . . . 11 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → ((𝑞(le‘𝐾)𝑋𝑟(le‘𝐾)𝑌) → (𝑝(le‘𝐾)(𝑞 𝑟) → 𝑝(le‘𝐾)(𝑋 𝑌))))
6261expimpd 628 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (((𝑞𝐵𝑟𝐵) ∧ (𝑞(le‘𝐾)𝑋𝑟(le‘𝐾)𝑌)) → (𝑝(le‘𝐾)(𝑞 𝑟) → 𝑝(le‘𝐾)(𝑋 𝑌))))
6346, 62sylani 687 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞(le‘𝐾)𝑋𝑟(le‘𝐾)𝑌)) → (𝑝(le‘𝐾)(𝑞 𝑟) → 𝑝(le‘𝐾)(𝑋 𝑌))))
6443, 63sylbid 230 . . . . . . . 8 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑞 ∈ (𝑀𝑋) ∧ 𝑟 ∈ (𝑀𝑌)) → (𝑝(le‘𝐾)(𝑞 𝑟) → 𝑝(le‘𝐾)(𝑋 𝑌))))
6564rexlimdvv 3066 . . . . . . 7 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (∃𝑞 ∈ (𝑀𝑋)∃𝑟 ∈ (𝑀𝑌)𝑝(le‘𝐾)(𝑞 𝑟) → 𝑝(le‘𝐾)(𝑋 𝑌)))
6665expimpd 628 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝𝐵 ∧ ∃𝑞 ∈ (𝑀𝑋)∃𝑟 ∈ (𝑀𝑌)𝑝(le‘𝐾)(𝑞 𝑟)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
675, 66sylani 687 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑀𝑋)∃𝑟 ∈ (𝑀𝑌)𝑝(le‘𝐾)(𝑞 𝑟)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
6834, 67jcad 554 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑀𝑋)∃𝑟 ∈ (𝑀𝑌)𝑝(le‘𝐾)(𝑞 𝑟)) → (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)(𝑋 𝑌))))
6932, 68jaod 394 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌)) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑀𝑋)∃𝑟 ∈ (𝑀𝑌)𝑝(le‘𝐾)(𝑞 𝑟))) → (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)(𝑋 𝑌))))
70 simp1 1081 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
713, 4, 35pmapssat 35363 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑀𝑋) ⊆ (Atoms‘𝐾))
72713adant3 1101 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑀𝑋) ⊆ (Atoms‘𝐾))
733, 4, 35pmapssat 35363 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑌𝐵) → (𝑀𝑌) ⊆ (Atoms‘𝐾))
74733adant2 1100 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑀𝑌) ⊆ (Atoms‘𝐾))
75 pmapjoin.p . . . . . 6 + = (+𝑃𝐾)
766, 7, 4, 75elpadd 35403 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑀𝑋) ⊆ (Atoms‘𝐾) ∧ (𝑀𝑌) ⊆ (Atoms‘𝐾)) → (𝑝 ∈ ((𝑀𝑋) + (𝑀𝑌)) ↔ ((𝑝 ∈ (𝑀𝑋) ∨ 𝑝 ∈ (𝑀𝑌)) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑀𝑋)∃𝑟 ∈ (𝑀𝑌)𝑝(le‘𝐾)(𝑞 𝑟)))))
7770, 72, 74, 76syl3anc 1366 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑝 ∈ ((𝑀𝑋) + (𝑀𝑌)) ↔ ((𝑝 ∈ (𝑀𝑋) ∨ 𝑝 ∈ (𝑀𝑌)) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑀𝑋)∃𝑟 ∈ (𝑀𝑌)𝑝(le‘𝐾)(𝑞 𝑟)))))
783, 6, 4, 35elpmap 35362 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑝 ∈ (𝑀𝑋) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋)))
79783adant3 1101 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑝 ∈ (𝑀𝑋) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋)))
803, 6, 4, 35elpmap 35362 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑌𝐵) → (𝑝 ∈ (𝑀𝑌) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌)))
81803adant2 1100 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑝 ∈ (𝑀𝑌) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌)))
8279, 81orbi12d 746 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (𝑀𝑋) ∨ 𝑝 ∈ (𝑀𝑌)) ↔ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌))))
8382orbi1d 739 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (((𝑝 ∈ (𝑀𝑋) ∨ 𝑝 ∈ (𝑀𝑌)) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑀𝑋)∃𝑟 ∈ (𝑀𝑌)𝑝(le‘𝐾)(𝑞 𝑟))) ↔ (((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌)) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑀𝑋)∃𝑟 ∈ (𝑀𝑌)𝑝(le‘𝐾)(𝑞 𝑟)))))
8477, 83bitrd 268 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑝 ∈ ((𝑀𝑋) + (𝑀𝑌)) ↔ (((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌)) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑀𝑋)∃𝑟 ∈ (𝑀𝑌)𝑝(le‘𝐾)(𝑞 𝑟)))))
853, 6, 4, 35elpmap 35362 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵) → (𝑝 ∈ (𝑀‘(𝑋 𝑌)) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)(𝑋 𝑌))))
8670, 13, 85syl2anc 694 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑝 ∈ (𝑀‘(𝑋 𝑌)) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)(𝑋 𝑌))))
8769, 84, 863imtr4d 283 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑝 ∈ ((𝑀𝑋) + (𝑀𝑌)) → 𝑝 ∈ (𝑀‘(𝑋 𝑌))))
8887ssrdv 3642 1 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑀𝑋) + (𝑀𝑌)) ⊆ (𝑀‘(𝑋 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wrex 2942  wss 3607   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  Latclat 17092  Atomscatm 34868  pmapcpmap 35101  +𝑃cpadd 35399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-poset 16993  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-lat 17093  df-ats 34872  df-pmap 35108  df-padd 35400
This theorem is referenced by:  pmapjat1  35457  hlmod1i  35460  paddunN  35531  pl42lem2N  35584
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