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Theorem osumcllem9N 34062
Description: Lemma for osumclN 34065. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
osumcllem.l = (le‘𝐾)
osumcllem.j = (join‘𝐾)
osumcllem.a 𝐴 = (Atoms‘𝐾)
osumcllem.p + = (+𝑃𝐾)
osumcllem.o = (⊥𝑃𝐾)
osumcllem.c 𝐶 = (PSubCl‘𝐾)
osumcllem.m 𝑀 = (𝑋 + {𝑝})
osumcllem.u 𝑈 = ( ‘( ‘(𝑋 + 𝑌)))
Assertion
Ref Expression
osumcllem9N (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀 = 𝑋)

Proof of Theorem osumcllem9N
StepHypRef Expression
1 inass 3785 . . . . . . 7 ((( 𝑋) ∩ 𝑈) ∩ 𝑀) = (( 𝑋) ∩ (𝑈𝑀))
2 simp11 1084 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝐾 ∈ HL)
3 simp13 1086 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑌𝐶)
4 simp21 1087 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋 ⊆ ( 𝑌))
5 osumcllem.l . . . . . . . . . 10 = (le‘𝐾)
6 osumcllem.j . . . . . . . . . 10 = (join‘𝐾)
7 osumcllem.a . . . . . . . . . 10 𝐴 = (Atoms‘𝐾)
8 osumcllem.p . . . . . . . . . 10 + = (+𝑃𝐾)
9 osumcllem.o . . . . . . . . . 10 = (⊥𝑃𝐾)
10 osumcllem.c . . . . . . . . . 10 𝐶 = (PSubCl‘𝐾)
11 osumcllem.m . . . . . . . . . 10 𝑀 = (𝑋 + {𝑝})
12 osumcllem.u . . . . . . . . . 10 𝑈 = ( ‘( ‘(𝑋 + 𝑌)))
135, 6, 7, 8, 9, 10, 11, 12osumcllem3N 34056 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑌𝐶𝑋 ⊆ ( 𝑌)) → (( 𝑋) ∩ 𝑈) = 𝑌)
142, 3, 4, 13syl3anc 1318 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (( 𝑋) ∩ 𝑈) = 𝑌)
1514ineq1d 3775 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ((( 𝑋) ∩ 𝑈) ∩ 𝑀) = (𝑌𝑀))
161, 15syl5eqr 2658 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (( 𝑋) ∩ (𝑈𝑀)) = (𝑌𝑀))
17 simp12 1085 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋𝐶)
187, 10psubclssatN 34039 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝐶) → 𝑋𝐴)
192, 17, 18syl2anc 691 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋𝐴)
207, 10psubclssatN 34039 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑌𝐶) → 𝑌𝐴)
212, 3, 20syl2anc 691 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑌𝐴)
22 simp22 1088 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋 ≠ ∅)
237, 8paddssat 33912 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) ⊆ 𝐴)
242, 19, 21, 23syl3anc 1318 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 + 𝑌) ⊆ 𝐴)
257, 9polssatN 34006 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ 𝐴) → ( ‘(𝑋 + 𝑌)) ⊆ 𝐴)
262, 24, 25syl2anc 691 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ‘(𝑋 + 𝑌)) ⊆ 𝐴)
277, 9polssatN 34006 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ ( ‘(𝑋 + 𝑌)) ⊆ 𝐴) → ( ‘( ‘(𝑋 + 𝑌))) ⊆ 𝐴)
282, 26, 27syl2anc 691 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ‘( ‘(𝑋 + 𝑌))) ⊆ 𝐴)
2912, 28syl5eqss 3612 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑈𝐴)
30 simp23 1089 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑝𝑈)
3129, 30sseldd 3569 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑝𝐴)
32 simp3 1056 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ¬ 𝑝 ∈ (𝑋 + 𝑌))
335, 6, 7, 8, 9, 10, 11, 12osumcllem8N 34061 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝐴) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑌𝑀) = ∅)
342, 19, 21, 4, 22, 31, 32, 33syl331anc 1343 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑌𝑀) = ∅)
3516, 34eqtrd 2644 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (( 𝑋) ∩ (𝑈𝑀)) = ∅)
3635fveq2d 6092 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ‘(( 𝑋) ∩ (𝑈𝑀))) = ( ‘∅))
377, 9pol0N 34007 . . . . 5 (𝐾 ∈ HL → ( ‘∅) = 𝐴)
382, 37syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ‘∅) = 𝐴)
3936, 38eqtrd 2644 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ‘(( 𝑋) ∩ (𝑈𝑀))) = 𝐴)
405, 6, 7, 8, 9, 10, 11, 12osumcllem1N 34054 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ 𝑝𝑈) → (𝑈𝑀) = 𝑀)
412, 19, 21, 30, 40syl31anc 1321 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑈𝑀) = 𝑀)
4239, 41ineq12d 3777 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (( ‘(( 𝑋) ∩ (𝑈𝑀))) ∩ (𝑈𝑀)) = (𝐴𝑀))
437, 9, 10polsubclN 34050 . . . . . 6 ((𝐾 ∈ HL ∧ ( ‘(𝑋 + 𝑌)) ⊆ 𝐴) → ( ‘( ‘(𝑋 + 𝑌))) ∈ 𝐶)
442, 26, 43syl2anc 691 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ‘( ‘(𝑋 + 𝑌))) ∈ 𝐶)
4512, 44syl5eqel 2692 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑈𝐶)
467, 8, 10paddatclN 34047 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐶𝑝𝐴) → (𝑋 + {𝑝}) ∈ 𝐶)
472, 17, 31, 46syl3anc 1318 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 + {𝑝}) ∈ 𝐶)
4811, 47syl5eqel 2692 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀𝐶)
4910psubclinN 34046 . . . 4 ((𝐾 ∈ HL ∧ 𝑈𝐶𝑀𝐶) → (𝑈𝑀) ∈ 𝐶)
502, 45, 48, 49syl3anc 1318 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑈𝑀) ∈ 𝐶)
515, 6, 7, 8, 9, 10, 11, 12osumcllem2N 34055 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ 𝑝𝑈) → 𝑋 ⊆ (𝑈𝑀))
522, 19, 21, 30, 51syl31anc 1321 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋 ⊆ (𝑈𝑀))
5310, 9poml6N 34053 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶 ∧ (𝑈𝑀) ∈ 𝐶) ∧ 𝑋 ⊆ (𝑈𝑀)) → (( ‘(( 𝑋) ∩ (𝑈𝑀))) ∩ (𝑈𝑀)) = 𝑋)
542, 17, 50, 52, 53syl31anc 1321 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (( ‘(( 𝑋) ∩ (𝑈𝑀))) ∩ (𝑈𝑀)) = 𝑋)
5531snssd 4281 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → {𝑝} ⊆ 𝐴)
567, 8paddssat 33912 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴 ∧ {𝑝} ⊆ 𝐴) → (𝑋 + {𝑝}) ⊆ 𝐴)
572, 19, 55, 56syl3anc 1318 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 + {𝑝}) ⊆ 𝐴)
5811, 57syl5eqss 3612 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀𝐴)
59 sseqin2 3779 . . 3 (𝑀𝐴 ↔ (𝐴𝑀) = 𝑀)
6058, 59sylib 207 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝐴𝑀) = 𝑀)
6142, 54, 603eqtr3rd 2653 1 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  w3a 1031   = wceq 1475  wcel 1977  wne 2780  cin 3539  wss 3540  c0 3874  {csn 4125  cfv 5790  (class class class)co 6527  lecple 15724  joincjn 16716  Atomscatm 33362  HLchlt 33449  +𝑃cpadd 33893  𝑃cpolN 34000  PSubClcpscN 34032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-riotaBAD 33051
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-iin 4453  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7037  df-2nd 7038  df-undef 7264  df-preset 16700  df-poset 16718  df-plt 16730  df-lub 16746  df-glb 16747  df-join 16748  df-meet 16749  df-p0 16811  df-p1 16812  df-lat 16818  df-clat 16880  df-oposet 33275  df-ol 33277  df-oml 33278  df-covers 33365  df-ats 33366  df-atl 33397  df-cvlat 33421  df-hlat 33450  df-psubsp 33601  df-pmap 33602  df-padd 33894  df-polarityN 34001  df-psubclN 34033
This theorem is referenced by:  osumcllem11N  34064
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