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Theorem pmapjoin 37145
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 486 . . . . . . 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 2798 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
53, 4atbase 36582 . . . . . . 7 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
6 eqid 2798 . . . . . . . . . . 11 (le‘𝐾) = (le‘𝐾)
7 pmapjoin.j . . . . . . . . . . 11 = (join‘𝐾)
83, 6, 7latlej1 17662 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
98adantr 484 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
10 simpl1 1188 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝐾 ∈ Lat)
11 simpr 488 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑝𝐵)
12 simpl2 1189 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑋𝐵)
133, 7latjcl 17653 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
1413adantr 484 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑋 𝑌) ∈ 𝐵)
153, 6lattr 17658 . . . . . . . . . 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 1369 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑝(le‘𝐾)𝑋𝑋(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
179, 16mpan2d 693 . . . . . . . 8 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)(𝑋 𝑌)))
1817expimpd 457 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝𝐵𝑝(le‘𝐾)𝑋) → 𝑝(le‘𝐾)(𝑋 𝑌)))
195, 18sylani 606 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) → 𝑝(le‘𝐾)(𝑋 𝑌)))
202, 19jcad 516 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) → (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)(𝑋 𝑌))))
21 simpl 486 . . . . . . 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 17663 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑌(le‘𝐾)(𝑋 𝑌))
2423adantr 484 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑌(le‘𝐾)(𝑋 𝑌))
25 simpl3 1190 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑌𝐵)
263, 6lattr 17658 . . . . . . . . . 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 1369 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑝(le‘𝐾)𝑌𝑌(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
2824, 27mpan2d 693 . . . . . . . 8 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑝(le‘𝐾)𝑌𝑝(le‘𝐾)(𝑋 𝑌)))
2928expimpd 457 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝𝐵𝑝(le‘𝐾)𝑌) → 𝑝(le‘𝐾)(𝑋 𝑌)))
305, 29sylani 606 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌) → 𝑝(le‘𝐾)(𝑋 𝑌)))
3122, 30jcad 516 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌) → (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)(𝑋 𝑌))))
3220, 31jaod 856 . . . 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 486 . . . . . 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 37051 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑞 ∈ (𝑀𝑋) ↔ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋)))
37363adant3 1129 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑞 ∈ (𝑀𝑋) ↔ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋)))
383, 6, 4, 35elpmap 37051 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑌𝐵) → (𝑟 ∈ (𝑀𝑌) ↔ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑌)))
39383adant2 1128 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑟 ∈ (𝑀𝑌) ↔ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑌)))
4037, 39anbi12d 633 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑞 ∈ (𝑀𝑋) ∧ 𝑟 ∈ (𝑀𝑌)) ↔ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑌))))
41 an4 655 . . . . . . . . . . 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 290 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑞 ∈ (𝑀𝑋) ∧ 𝑟 ∈ (𝑀𝑌)) ↔ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞(le‘𝐾)𝑋𝑟(le‘𝐾)𝑌))))
4342adantr 484 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑞 ∈ (𝑀𝑋) ∧ 𝑟 ∈ (𝑀𝑌)) ↔ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞(le‘𝐾)𝑋𝑟(le‘𝐾)𝑌))))
443, 4atbase 36582 . . . . . . . . . . 11 (𝑞 ∈ (Atoms‘𝐾) → 𝑞𝐵)
453, 4atbase 36582 . . . . . . . . . . 11 (𝑟 ∈ (Atoms‘𝐾) → 𝑟𝐵)
4644, 45anim12i 615 . . . . . . . . . 10 ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑞𝐵𝑟𝐵))
47 simpll1 1209 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝐾 ∈ Lat)
48 simprl 770 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑞𝐵)
49 simpll2 1210 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑋𝐵)
50 simprr 772 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑟𝐵)
51 simpll3 1211 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑌𝐵)
523, 6, 7latjlej12 17669 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ (𝑞𝐵𝑋𝐵) ∧ (𝑟𝐵𝑌𝐵)) → ((𝑞(le‘𝐾)𝑋𝑟(le‘𝐾)𝑌) → (𝑞 𝑟)(le‘𝐾)(𝑋 𝑌)))
5347, 48, 49, 50, 51, 52syl122anc 1376 . . . . . . . . . . . 12 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → ((𝑞(le‘𝐾)𝑋𝑟(le‘𝐾)𝑌) → (𝑞 𝑟)(le‘𝐾)(𝑋 𝑌)))
54 simplr 768 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑝𝐵)
553, 7latjcl 17653 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ 𝑞𝐵𝑟𝐵) → (𝑞 𝑟) ∈ 𝐵)
5647, 48, 50, 55syl3anc 1368 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → (𝑞 𝑟) ∈ 𝐵)
5713ad2antrr 725 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → (𝑋 𝑌) ∈ 𝐵)
583, 6lattr 17658 . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → ((𝑝(le‘𝐾)(𝑞 𝑟) ∧ (𝑞 𝑟)(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
6059expcomd 420 . . . . . . . . . . . 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 457 . . . . . . . . . 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 606 . . . . . . . . 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 243 . . . . . . . 8 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑞 ∈ (𝑀𝑋) ∧ 𝑟 ∈ (𝑀𝑌)) → (𝑝(le‘𝐾)(𝑞 𝑟) → 𝑝(le‘𝐾)(𝑋 𝑌))))
6564rexlimdvv 3252 . . . . . . 7 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (∃𝑞 ∈ (𝑀𝑋)∃𝑟 ∈ (𝑀𝑌)𝑝(le‘𝐾)(𝑞 𝑟) → 𝑝(le‘𝐾)(𝑋 𝑌)))
6665expimpd 457 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝𝐵 ∧ ∃𝑞 ∈ (𝑀𝑋)∃𝑟 ∈ (𝑀𝑌)𝑝(le‘𝐾)(𝑞 𝑟)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
675, 66sylani 606 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑀𝑋)∃𝑟 ∈ (𝑀𝑌)𝑝(le‘𝐾)(𝑞 𝑟)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
6834, 67jcad 516 . . . 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 856 . . 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 1133 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
713, 4, 35pmapssat 37052 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑀𝑋) ⊆ (Atoms‘𝐾))
72713adant3 1129 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑀𝑋) ⊆ (Atoms‘𝐾))
733, 4, 35pmapssat 37052 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑌𝐵) → (𝑀𝑌) ⊆ (Atoms‘𝐾))
74733adant2 1128 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑀𝑌) ⊆ (Atoms‘𝐾))
75 pmapjoin.p . . . . . 6 + = (+𝑃𝐾)
766, 7, 4, 75elpadd 37092 . . . . 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 1368 . . . 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 37051 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑝 ∈ (𝑀𝑋) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋)))
79783adant3 1129 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑝 ∈ (𝑀𝑋) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋)))
803, 6, 4, 35elpmap 37051 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑌𝐵) → (𝑝 ∈ (𝑀𝑌) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌)))
81803adant2 1128 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑝 ∈ (𝑀𝑌) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌)))
8279, 81orbi12d 916 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (𝑀𝑋) ∨ 𝑝 ∈ (𝑀𝑌)) ↔ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌))))
8382orbi1d 914 . . . 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 282 . . 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 37051 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵) → (𝑝 ∈ (𝑀‘(𝑋 𝑌)) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)(𝑋 𝑌))))
8670, 13, 85syl2anc 587 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑝 ∈ (𝑀‘(𝑋 𝑌)) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)(𝑋 𝑌))))
8769, 84, 863imtr4d 297 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑝 ∈ ((𝑀𝑋) + (𝑀𝑌)) → 𝑝 ∈ (𝑀‘(𝑋 𝑌))))
8887ssrdv 3921 1 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑀𝑋) + (𝑀𝑌)) ⊆ (𝑀‘(𝑋 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2111  wrex 3107  wss 3881   class class class wbr 5030  cfv 6324  (class class class)co 7135  Basecbs 16475  lecple 16564  joincjn 17546  Latclat 17647  Atomscatm 36556  pmapcpmap 36790  +𝑃cpadd 37088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-poset 17548  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-lat 17648  df-ats 36560  df-pmap 36797  df-padd 37089
This theorem is referenced by:  pmapjat1  37146  hlmod1i  37149  paddunN  37220  pl42lem2N  37273
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