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 39499
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 481 . . . . . . 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 2725 . . . . . . . 8 (Atoms‘𝐾) = (Atoms‘𝐾)
53, 4atbase 38935 . . . . . . 7 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
6 eqid 2725 . . . . . . . . . . 11 (le‘𝐾) = (le‘𝐾)
7 pmapjoin.j . . . . . . . . . . 11 = (join‘𝐾)
83, 6, 7latlej1 18468 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
98adantr 479 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
10 simpl1 1188 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝐾 ∈ Lat)
11 simpr 483 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑝𝐵)
12 simpl2 1189 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑋𝐵)
133, 7latjcl 18459 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
1413adantr 479 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑋 𝑌) ∈ 𝐵)
153, 6lattr 18464 . . . . . . . . . 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 692 . . . . . . . 8 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑝(le‘𝐾)𝑋𝑝(le‘𝐾)(𝑋 𝑌)))
1817expimpd 452 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝𝐵𝑝(le‘𝐾)𝑋) → 𝑝(le‘𝐾)(𝑋 𝑌)))
195, 18sylani 602 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) → 𝑝(le‘𝐾)(𝑋 𝑌)))
202, 19jcad 511 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) → (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)(𝑋 𝑌))))
21 simpl 481 . . . . . . 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 18469 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑌(le‘𝐾)(𝑋 𝑌))
2423adantr 479 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑌(le‘𝐾)(𝑋 𝑌))
25 simpl3 1190 . . . . . . . . . 10 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → 𝑌𝐵)
263, 6lattr 18464 . . . . . . . . . 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 692 . . . . . . . 8 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → (𝑝(le‘𝐾)𝑌𝑝(le‘𝐾)(𝑋 𝑌)))
2928expimpd 452 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝𝐵𝑝(le‘𝐾)𝑌) → 𝑝(le‘𝐾)(𝑋 𝑌)))
305, 29sylani 602 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑌) → 𝑝(le‘𝐾)(𝑋 𝑌)))
3122, 30jcad 511 . . . . 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 481 . . . . . 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 39405 . . . . . . . . . . . . 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 39405 . . . . . . . . . . . . 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 630 . . . . . . . . . . 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 479 . . . . . . . . 9 (((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) → ((𝑞 ∈ (𝑀𝑋) ∧ 𝑟 ∈ (𝑀𝑌)) ↔ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞(le‘𝐾)𝑋𝑟(le‘𝐾)𝑌))))
443, 4atbase 38935 . . . . . . . . . . 11 (𝑞 ∈ (Atoms‘𝐾) → 𝑞𝐵)
453, 4atbase 38935 . . . . . . . . . . 11 (𝑟 ∈ (Atoms‘𝐾) → 𝑟𝐵)
4644, 45anim12i 611 . . . . . . . . . 10 ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑞𝐵𝑟𝐵))
47 simpll1 1209 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝐾 ∈ Lat)
48 simprl 769 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑞𝐵)
49 simpll2 1210 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑋𝐵)
50 simprr 771 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑟𝐵)
51 simpll3 1211 . . . . . . . . . . . . 13 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑌𝐵)
523, 6, 7latjlej12 18475 . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → 𝑝𝐵)
553, 7latjcl 18459 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ 𝑞𝐵𝑟𝐵) → (𝑞 𝑟) ∈ 𝐵)
5647, 48, 50, 55syl3anc 1368 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → (𝑞 𝑟) ∈ 𝐵)
5713ad2antrr 724 . . . . . . . . . . . . . 14 ((((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑝𝐵) ∧ (𝑞𝐵𝑟𝐵)) → (𝑋 𝑌) ∈ 𝐵)
583, 6lattr 18464 . . . . . . . . . . . . . 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 415 . . . . . . . . . . . 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 452 . . . . . . . . . 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 602 . . . . . . . . 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 452 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝𝐵 ∧ ∃𝑞 ∈ (𝑀𝑋)∃𝑟 ∈ (𝑀𝑌)𝑝(le‘𝐾)(𝑞 𝑟)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
675, 66sylani 602 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑝 ∈ (Atoms‘𝐾) ∧ ∃𝑞 ∈ (𝑀𝑋)∃𝑟 ∈ (𝑀𝑌)𝑝(le‘𝐾)(𝑞 𝑟)) → 𝑝(le‘𝐾)(𝑋 𝑌)))
6834, 67jcad 511 . . . 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 1133 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
713, 4, 35pmapssat 39406 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑀𝑋) ⊆ (Atoms‘𝐾))
72713adant3 1129 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑀𝑋) ⊆ (Atoms‘𝐾))
733, 4, 35pmapssat 39406 . . . . . 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 39446 . . . . 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 39405 . . . . . . 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 39405 . . . . . . 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 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 39405 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵) → (𝑝 ∈ (𝑀‘(𝑋 𝑌)) ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)(𝑋 𝑌))))
8670, 13, 85syl2anc 582 . . 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 3984 1 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑀𝑋) + (𝑀𝑌)) ⊆ (𝑀‘(𝑋 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 845  w3a 1084   = wceq 1533  wcel 2098  wrex 3059  wss 3946   class class class wbr 5152  cfv 6553  (class class class)co 7423  Basecbs 17208  lecple 17268  joincjn 18331  Latclat 18451  Atomscatm 38909  pmapcpmap 39144  +𝑃cpadd 39442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7379  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 8002  df-2nd 8003  df-poset 18333  df-lub 18366  df-glb 18367  df-join 18368  df-meet 18369  df-lat 18452  df-ats 38913  df-pmap 39151  df-padd 39443
This theorem is referenced by:  pmapjat1  39500  hlmod1i  39503  paddunN  39574  pl42lem2N  39627
  Copyright terms: Public domain W3C validator