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Theorem osumcllem9N 35568
Description: Lemma for osumclN 35571. (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 3856 . . . . . . 7 ((( 𝑋) ∩ 𝑈) ∩ 𝑀) = (( 𝑋) ∩ (𝑈𝑀))
2 simp11 1111 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝐾 ∈ HL)
3 simp13 1113 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑌𝐶)
4 simp21 1114 . . . . . . . . 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 35562 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑌𝐶𝑋 ⊆ ( 𝑌)) → (( 𝑋) ∩ 𝑈) = 𝑌)
142, 3, 4, 13syl3anc 1366 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (( 𝑋) ∩ 𝑈) = 𝑌)
1514ineq1d 3846 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ((( 𝑋) ∩ 𝑈) ∩ 𝑀) = (𝑌𝑀))
161, 15syl5eqr 2699 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (( 𝑋) ∩ (𝑈𝑀)) = (𝑌𝑀))
17 simp12 1112 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋𝐶)
187, 10psubclssatN 35545 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝐶) → 𝑋𝐴)
192, 17, 18syl2anc 694 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋𝐴)
207, 10psubclssatN 35545 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑌𝐶) → 𝑌𝐴)
212, 3, 20syl2anc 694 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑌𝐴)
22 simp22 1115 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋 ≠ ∅)
237, 8paddssat 35418 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) ⊆ 𝐴)
242, 19, 21, 23syl3anc 1366 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 + 𝑌) ⊆ 𝐴)
257, 9polssatN 35512 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ 𝐴) → ( ‘(𝑋 + 𝑌)) ⊆ 𝐴)
262, 24, 25syl2anc 694 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ‘(𝑋 + 𝑌)) ⊆ 𝐴)
277, 9polssatN 35512 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ ( ‘(𝑋 + 𝑌)) ⊆ 𝐴) → ( ‘( ‘(𝑋 + 𝑌))) ⊆ 𝐴)
282, 26, 27syl2anc 694 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ‘( ‘(𝑋 + 𝑌))) ⊆ 𝐴)
2912, 28syl5eqss 3682 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑈𝐴)
30 simp23 1116 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑝𝑈)
3129, 30sseldd 3637 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑝𝐴)
32 simp3 1083 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ¬ 𝑝 ∈ (𝑋 + 𝑌))
335, 6, 7, 8, 9, 10, 11, 12osumcllem8N 35567 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝐴) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑌𝑀) = ∅)
342, 19, 21, 4, 22, 31, 32, 33syl331anc 1391 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑌𝑀) = ∅)
3516, 34eqtrd 2685 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (( 𝑋) ∩ (𝑈𝑀)) = ∅)
3635fveq2d 6233 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ‘(( 𝑋) ∩ (𝑈𝑀))) = ( ‘∅))
377, 9pol0N 35513 . . . . 5 (𝐾 ∈ HL → ( ‘∅) = 𝐴)
382, 37syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ‘∅) = 𝐴)
3936, 38eqtrd 2685 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ‘(( 𝑋) ∩ (𝑈𝑀))) = 𝐴)
405, 6, 7, 8, 9, 10, 11, 12osumcllem1N 35560 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ 𝑝𝑈) → (𝑈𝑀) = 𝑀)
412, 19, 21, 30, 40syl31anc 1369 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑈𝑀) = 𝑀)
4239, 41ineq12d 3848 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (( ‘(( 𝑋) ∩ (𝑈𝑀))) ∩ (𝑈𝑀)) = (𝐴𝑀))
437, 9, 10polsubclN 35556 . . . . . 6 ((𝐾 ∈ HL ∧ ( ‘(𝑋 + 𝑌)) ⊆ 𝐴) → ( ‘( ‘(𝑋 + 𝑌))) ∈ 𝐶)
442, 26, 43syl2anc 694 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ‘( ‘(𝑋 + 𝑌))) ∈ 𝐶)
4512, 44syl5eqel 2734 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑈𝐶)
467, 8, 10paddatclN 35553 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐶𝑝𝐴) → (𝑋 + {𝑝}) ∈ 𝐶)
472, 17, 31, 46syl3anc 1366 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 + {𝑝}) ∈ 𝐶)
4811, 47syl5eqel 2734 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀𝐶)
4910psubclinN 35552 . . . 4 ((𝐾 ∈ HL ∧ 𝑈𝐶𝑀𝐶) → (𝑈𝑀) ∈ 𝐶)
502, 45, 48, 49syl3anc 1366 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑈𝑀) ∈ 𝐶)
515, 6, 7, 8, 9, 10, 11, 12osumcllem2N 35561 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ 𝑝𝑈) → 𝑋 ⊆ (𝑈𝑀))
522, 19, 21, 30, 51syl31anc 1369 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋 ⊆ (𝑈𝑀))
5310, 9poml6N 35559 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶 ∧ (𝑈𝑀) ∈ 𝐶) ∧ 𝑋 ⊆ (𝑈𝑀)) → (( ‘(( 𝑋) ∩ (𝑈𝑀))) ∩ (𝑈𝑀)) = 𝑋)
542, 17, 50, 52, 53syl31anc 1369 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (( ‘(( 𝑋) ∩ (𝑈𝑀))) ∩ (𝑈𝑀)) = 𝑋)
5531snssd 4372 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → {𝑝} ⊆ 𝐴)
567, 8paddssat 35418 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴 ∧ {𝑝} ⊆ 𝐴) → (𝑋 + {𝑝}) ⊆ 𝐴)
572, 19, 55, 56syl3anc 1366 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 + {𝑝}) ⊆ 𝐴)
5811, 57syl5eqss 3682 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀𝐴)
59 sseqin2 3850 . . 3 (𝑀𝐴 ↔ (𝐴𝑀) = 𝑀)
6058, 59sylib 208 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝐴𝑀) = 𝑀)
6142, 54, 603eqtr3rd 2694 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 1054   = wceq 1523  wcel 2030  wne 2823  cin 3606  wss 3607  c0 3948  {csn 4210  cfv 5926  (class class class)co 6690  lecple 15995  joincjn 16991  Atomscatm 34868  HLchlt 34955  +𝑃cpadd 35399  𝑃cpolN 35506  PSubClcpscN 35538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-riotaBAD 34557
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-undef 7444  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-p1 17087  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-psubsp 35107  df-pmap 35108  df-padd 35400  df-polarityN 35507  df-psubclN 35539
This theorem is referenced by:  osumcllem11N  35570
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