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Theorem pmapjoin 38128
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 483 . . . . . . 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 2736 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
53, 4atbase 37564 . . . . . . 7 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
6 eqid 2736 . . . . . . . . . . 11 (le‘𝐾) = (le‘𝐾)
7 pmapjoin.j . . . . . . . . . . 11 = (join‘𝐾)
83, 6, 7latlej1 18263 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
98adantr 481 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
10 simpl1 1190 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝐾 ∈ Lat)
11 simpr 485 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑝𝐵)
12 simpl2 1191 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑋𝐵)
133, 7latjcl 18254 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
1413adantr 481 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑋 𝑌) ∈ 𝐵)
153, 6lattr 18259 . . . . . . . . . 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 1371 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑝(le‘𝐾)𝑋𝑋(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
179, 16mpan2d 691 . . . . . . . 8 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)(𝑋 𝑌)))
1817expimpd 454 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝𝐵𝑝(le‘𝐾)𝑋) → 𝑝(le‘𝐾)(𝑋 𝑌)))
195, 18sylani 604 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) → 𝑝(le‘𝐾)(𝑋 𝑌)))
202, 19jcad 513 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) → (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)(𝑋 𝑌))))
21 simpl 483 . . . . . . 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 18264 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑌(le‘𝐾)(𝑋 𝑌))
2423adantr 481 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑌(le‘𝐾)(𝑋 𝑌))
25 simpl3 1192 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑌𝐵)
263, 6lattr 18259 . . . . . . . . . 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 1371 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑝(le‘𝐾)𝑌𝑌(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
2824, 27mpan2d 691 . . . . . . . 8 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑝(le‘𝐾)𝑌𝑝(le‘𝐾)(𝑋 𝑌)))
2928expimpd 454 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝𝐵𝑝(le‘𝐾)𝑌) → 𝑝(le‘𝐾)(𝑋 𝑌)))
305, 29sylani 604 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌) → 𝑝(le‘𝐾)(𝑋 𝑌)))
3122, 30jcad 513 . . . . 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 483 . . . . . 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 38034 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑞 ∈ (𝑀𝑋) ↔ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋)))
37363adant3 1131 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑞 ∈ (𝑀𝑋) ↔ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋)))
383, 6, 4, 35elpmap 38034 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑌𝐵) → (𝑟 ∈ (𝑀𝑌) ↔ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑌)))
39383adant2 1130 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑟 ∈ (𝑀𝑌) ↔ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑌)))
4037, 39anbi12d 631 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑞 ∈ (𝑀𝑋) ∧ 𝑟 ∈ (𝑀𝑌)) ↔ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑌))))
41 an4 653 . . . . . . . . . . 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, 41bitrdi 286 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑞 ∈ (𝑀𝑋) ∧ 𝑟 ∈ (𝑀𝑌)) ↔ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞(le‘𝐾)𝑋𝑟(le‘𝐾)𝑌))))
4342adantr 481 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑞 ∈ (𝑀𝑋) ∧ 𝑟 ∈ (𝑀𝑌)) ↔ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞(le‘𝐾)𝑋𝑟(le‘𝐾)𝑌))))
443, 4atbase 37564 . . . . . . . . . . 11 (𝑞 ∈ (Atoms‘𝐾) → 𝑞𝐵)
453, 4atbase 37564 . . . . . . . . . . 11 (𝑟 ∈ (Atoms‘𝐾) → 𝑟𝐵)
4644, 45anim12i 613 . . . . . . . . . 10 ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑞𝐵𝑟𝐵))
47 simpll1 1211 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝐾 ∈ Lat)
48 simprl 768 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑞𝐵)
49 simpll2 1212 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑋𝐵)
50 simprr 770 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑟𝐵)
51 simpll3 1213 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑌𝐵)
523, 6, 7latjlej12 18270 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ (𝑞𝐵𝑋𝐵) ∧ (𝑟𝐵𝑌𝐵)) → ((𝑞(le‘𝐾)𝑋𝑟(le‘𝐾)𝑌) → (𝑞 𝑟)(le‘𝐾)(𝑋 𝑌)))
5347, 48, 49, 50, 51, 52syl122anc 1378 . . . . . . . . . . . 12 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → ((𝑞(le‘𝐾)𝑋𝑟(le‘𝐾)𝑌) → (𝑞 𝑟)(le‘𝐾)(𝑋 𝑌)))
54 simplr 766 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑝𝐵)
553, 7latjcl 18254 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ 𝑞𝐵𝑟𝐵) → (𝑞 𝑟) ∈ 𝐵)
5647, 48, 50, 55syl3anc 1370 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → (𝑞 𝑟) ∈ 𝐵)
5713ad2antrr 723 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → (𝑋 𝑌) ∈ 𝐵)
583, 6lattr 18259 . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → ((𝑝(le‘𝐾)(𝑞 𝑟) ∧ (𝑞 𝑟)(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
6059expcomd 417 . . . . . . . . . . . 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 454 . . . . . . . . . 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 604 . . . . . . . . 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 239 . . . . . . . 8 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑞 ∈ (𝑀𝑋) ∧ 𝑟 ∈ (𝑀𝑌)) → (𝑝(le‘𝐾)(𝑞 𝑟) → 𝑝(le‘𝐾)(𝑋 𝑌))))
6564rexlimdvv 3200 . . . . . . 7 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (∃𝑞 ∈ (𝑀𝑋)∃𝑟 ∈ (𝑀𝑌)𝑝(le‘𝐾)(𝑞 𝑟) → 𝑝(le‘𝐾)(𝑋 𝑌)))
6665expimpd 454 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝𝐵 ∧ ∃𝑞 ∈ (𝑀𝑋)∃𝑟 ∈ (𝑀𝑌)𝑝(le‘𝐾)(𝑞 𝑟)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
675, 66sylani 604 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑀𝑋)∃𝑟 ∈ (𝑀𝑌)𝑝(le‘𝐾)(𝑞 𝑟)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
6834, 67jcad 513 . . . 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 1135 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
713, 4, 35pmapssat 38035 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑀𝑋) ⊆ (Atoms‘𝐾))
72713adant3 1131 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑀𝑋) ⊆ (Atoms‘𝐾))
733, 4, 35pmapssat 38035 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑌𝐵) → (𝑀𝑌) ⊆ (Atoms‘𝐾))
74733adant2 1130 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑀𝑌) ⊆ (Atoms‘𝐾))
75 pmapjoin.p . . . . . 6 + = (+𝑃𝐾)
766, 7, 4, 75elpadd 38075 . . . . 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 1370 . . . 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 38034 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑝 ∈ (𝑀𝑋) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋)))
79783adant3 1131 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑝 ∈ (𝑀𝑋) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋)))
803, 6, 4, 35elpmap 38034 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑌𝐵) → (𝑝 ∈ (𝑀𝑌) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌)))
81803adant2 1130 . . . . . 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 278 . . 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 38034 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵) → (𝑝 ∈ (𝑀‘(𝑋 𝑌)) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)(𝑋 𝑌))))
8670, 13, 85syl2anc 584 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑝 ∈ (𝑀‘(𝑋 𝑌)) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)(𝑋 𝑌))))
8769, 84, 863imtr4d 293 . 2 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑝 ∈ ((𝑀𝑋) + (𝑀𝑌)) → 𝑝 ∈ (𝑀‘(𝑋 𝑌))))
8887ssrdv 3938 1 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑀𝑋) + (𝑀𝑌)) ⊆ (𝑀‘(𝑋 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 844  w3a 1086   = wceq 1540  wcel 2105  wrex 3070  wss 3898   class class class wbr 5092  cfv 6479  (class class class)co 7337  Basecbs 17009  lecple 17066  joincjn 18126  Latclat 18246  Atomscatm 37538  pmapcpmap 37773  +𝑃cpadd 38071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-1st 7899  df-2nd 7900  df-poset 18128  df-lub 18161  df-glb 18162  df-join 18163  df-meet 18164  df-lat 18247  df-ats 37542  df-pmap 37780  df-padd 38072
This theorem is referenced by:  pmapjat1  38129  hlmod1i  38132  paddunN  38203  pl42lem2N  38256
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