Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pmapjoin Structured version   Visualization version   GIF version

Theorem pmapjoin 38315
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 37751 . . . . . . 7 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
6 eqid 2736 . . . . . . . . . . 11 (le‘𝐾) = (le‘𝐾)
7 pmapjoin.j . . . . . . . . . . 11 = (join‘𝐾)
83, 6, 7latlej1 18337 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
98adantr 481 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
10 simpl1 1191 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝐾 ∈ Lat)
11 simpr 485 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑝𝐵)
12 simpl2 1192 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑋𝐵)
133, 7latjcl 18328 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
1413adantr 481 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑋 𝑌) ∈ 𝐵)
153, 6lattr 18333 . . . . . . . . . 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 1372 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑝(le‘𝐾)𝑋𝑋(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
179, 16mpan2d 692 . . . . . . . 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 18338 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑌(le‘𝐾)(𝑋 𝑌))
2423adantr 481 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑌(le‘𝐾)(𝑋 𝑌))
25 simpl3 1193 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑌𝐵)
263, 6lattr 18333 . . . . . . . . . 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 1372 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑝(le‘𝐾)𝑌𝑌(le‘𝐾)(𝑋 𝑌)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
2824, 27mpan2d 692 . . . . . . . 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 857 . . . 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 38221 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑞 ∈ (𝑀𝑋) ↔ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋)))
37363adant3 1132 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑞 ∈ (𝑀𝑋) ↔ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋)))
383, 6, 4, 35elpmap 38221 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑌𝐵) → (𝑟 ∈ (𝑀𝑌) ↔ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑌)))
39383adant2 1131 . . . . . . . . . . . 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 654 . . . . . . . . . . 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 37751 . . . . . . . . . . 11 (𝑞 ∈ (Atoms‘𝐾) → 𝑞𝐵)
453, 4atbase 37751 . . . . . . . . . . 11 (𝑟 ∈ (Atoms‘𝐾) → 𝑟𝐵)
4644, 45anim12i 613 . . . . . . . . . 10 ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑞𝐵𝑟𝐵))
47 simpll1 1212 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝐾 ∈ Lat)
48 simprl 769 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑞𝐵)
49 simpll2 1213 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑋𝐵)
50 simprr 771 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑟𝐵)
51 simpll3 1214 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑌𝐵)
523, 6, 7latjlej12 18344 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ (𝑞𝐵𝑋𝐵) ∧ (𝑟𝐵𝑌𝐵)) → ((𝑞(le‘𝐾)𝑋𝑟(le‘𝐾)𝑌) → (𝑞 𝑟)(le‘𝐾)(𝑋 𝑌)))
5347, 48, 49, 50, 51, 52syl122anc 1379 . . . . . . . . . . . 12 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → ((𝑞(le‘𝐾)𝑋𝑟(le‘𝐾)𝑌) → (𝑞 𝑟)(le‘𝐾)(𝑋 𝑌)))
54 simplr 767 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑝𝐵)
553, 7latjcl 18328 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ 𝑞𝐵𝑟𝐵) → (𝑞 𝑟) ∈ 𝐵)
5647, 48, 50, 55syl3anc 1371 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → (𝑞 𝑟) ∈ 𝐵)
5713ad2antrr 724 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → (𝑋 𝑌) ∈ 𝐵)
583, 6lattr 18333 . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . 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 3204 . . . . . . 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 857 . . 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 1136 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
713, 4, 35pmapssat 38222 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑀𝑋) ⊆ (Atoms‘𝐾))
72713adant3 1132 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑀𝑋) ⊆ (Atoms‘𝐾))
733, 4, 35pmapssat 38222 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑌𝐵) → (𝑀𝑌) ⊆ (Atoms‘𝐾))
74733adant2 1131 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑀𝑌) ⊆ (Atoms‘𝐾))
75 pmapjoin.p . . . . . 6 + = (+𝑃𝐾)
766, 7, 4, 75elpadd 38262 . . . . 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 1371 . . . 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 38221 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑝 ∈ (𝑀𝑋) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋)))
79783adant3 1132 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑝 ∈ (𝑀𝑋) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋)))
803, 6, 4, 35elpmap 38221 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑌𝐵) → (𝑝 ∈ (𝑀𝑌) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌)))
81803adant2 1131 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑝 ∈ (𝑀𝑌) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌)))
8279, 81orbi12d 917 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (𝑀𝑋) ∨ 𝑝 ∈ (𝑀𝑌)) ↔ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) ∨ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌))))
8382orbi1d 915 . . . 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 38221 . . . 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 3950 1 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑀𝑋) + (𝑀𝑌)) ⊆ (𝑀‘(𝑋 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wrex 3073  wss 3910   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  Latclat 18320  Atomscatm 37725  pmapcpmap 37960  +𝑃cpadd 38258
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-poset 18202  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-lat 18321  df-ats 37729  df-pmap 37967  df-padd 38259
This theorem is referenced by:  pmapjat1  38316  hlmod1i  38319  paddunN  38390  pl42lem2N  38443
  Copyright terms: Public domain W3C validator