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Theorem osumcllem9N 37092
Description: Lemma for osumclN 37095. (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 4194 . . . . . . 7 ((( 𝑋) ∩ 𝑈) ∩ 𝑀) = (( 𝑋) ∩ (𝑈𝑀))
2 simp11 1197 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝐾 ∈ HL)
3 simp13 1199 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑌𝐶)
4 simp21 1200 . . . . . . . . 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 37086 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑌𝐶𝑋 ⊆ ( 𝑌)) → (( 𝑋) ∩ 𝑈) = 𝑌)
142, 3, 4, 13syl3anc 1365 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (( 𝑋) ∩ 𝑈) = 𝑌)
1514ineq1d 4186 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ((( 𝑋) ∩ 𝑈) ∩ 𝑀) = (𝑌𝑀))
161, 15syl5eqr 2868 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (( 𝑋) ∩ (𝑈𝑀)) = (𝑌𝑀))
17 simp12 1198 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋𝐶)
187, 10psubclssatN 37069 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋𝐶) → 𝑋𝐴)
192, 17, 18syl2anc 586 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋𝐴)
207, 10psubclssatN 37069 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑌𝐶) → 𝑌𝐴)
212, 3, 20syl2anc 586 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑌𝐴)
22 simp22 1201 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋 ≠ ∅)
237, 8paddssat 36942 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) ⊆ 𝐴)
242, 19, 21, 23syl3anc 1365 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 + 𝑌) ⊆ 𝐴)
257, 9polssatN 37036 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ 𝐴) → ( ‘(𝑋 + 𝑌)) ⊆ 𝐴)
262, 24, 25syl2anc 586 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ‘(𝑋 + 𝑌)) ⊆ 𝐴)
277, 9polssatN 37036 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ ( ‘(𝑋 + 𝑌)) ⊆ 𝐴) → ( ‘( ‘(𝑋 + 𝑌))) ⊆ 𝐴)
282, 26, 27syl2anc 586 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ‘( ‘(𝑋 + 𝑌))) ⊆ 𝐴)
2912, 28eqsstrid 4013 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑈𝐴)
30 simp23 1202 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑝𝑈)
3129, 30sseldd 3966 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑝𝐴)
32 simp3 1132 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ¬ 𝑝 ∈ (𝑋 + 𝑌))
335, 6, 7, 8, 9, 10, 11, 12osumcllem8N 37091 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝐴) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑌𝑀) = ∅)
342, 19, 21, 4, 22, 31, 32, 33syl331anc 1389 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑌𝑀) = ∅)
3516, 34eqtrd 2854 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (( 𝑋) ∩ (𝑈𝑀)) = ∅)
3635fveq2d 6667 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ‘(( 𝑋) ∩ (𝑈𝑀))) = ( ‘∅))
377, 9pol0N 37037 . . . . 5 (𝐾 ∈ HL → ( ‘∅) = 𝐴)
382, 37syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ‘∅) = 𝐴)
3936, 38eqtrd 2854 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ‘(( 𝑋) ∩ (𝑈𝑀))) = 𝐴)
405, 6, 7, 8, 9, 10, 11, 12osumcllem1N 37084 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ 𝑝𝑈) → (𝑈𝑀) = 𝑀)
412, 19, 21, 30, 40syl31anc 1367 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑈𝑀) = 𝑀)
4239, 41ineq12d 4188 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (( ‘(( 𝑋) ∩ (𝑈𝑀))) ∩ (𝑈𝑀)) = (𝐴𝑀))
437, 9, 10polsubclN 37080 . . . . . 6 ((𝐾 ∈ HL ∧ ( ‘(𝑋 + 𝑌)) ⊆ 𝐴) → ( ‘( ‘(𝑋 + 𝑌))) ∈ 𝐶)
442, 26, 43syl2anc 586 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ( ‘( ‘(𝑋 + 𝑌))) ∈ 𝐶)
4512, 44eqeltrid 2915 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑈𝐶)
467, 8, 10paddatclN 37077 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝐶𝑝𝐴) → (𝑋 + {𝑝}) ∈ 𝐶)
472, 17, 31, 46syl3anc 1365 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 + {𝑝}) ∈ 𝐶)
4811, 47eqeltrid 2915 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀𝐶)
4910psubclinN 37076 . . . 4 ((𝐾 ∈ HL ∧ 𝑈𝐶𝑀𝐶) → (𝑈𝑀) ∈ 𝐶)
502, 45, 48, 49syl3anc 1365 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑈𝑀) ∈ 𝐶)
515, 6, 7, 8, 9, 10, 11, 12osumcllem2N 37085 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ 𝑝𝑈) → 𝑋 ⊆ (𝑈𝑀))
522, 19, 21, 30, 51syl31anc 1367 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋 ⊆ (𝑈𝑀))
5310, 9poml6N 37083 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶 ∧ (𝑈𝑀) ∈ 𝐶) ∧ 𝑋 ⊆ (𝑈𝑀)) → (( ‘(( 𝑋) ∩ (𝑈𝑀))) ∩ (𝑈𝑀)) = 𝑋)
542, 17, 50, 52, 53syl31anc 1367 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (( ‘(( 𝑋) ∩ (𝑈𝑀))) ∩ (𝑈𝑀)) = 𝑋)
5531snssd 4734 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → {𝑝} ⊆ 𝐴)
567, 8paddssat 36942 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴 ∧ {𝑝} ⊆ 𝐴) → (𝑋 + {𝑝}) ⊆ 𝐴)
572, 19, 55, 56syl3anc 1365 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑋 + {𝑝}) ⊆ 𝐴)
5811, 57eqsstrid 4013 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀𝐴)
59 sseqin2 4190 . . 3 (𝑀𝐴 ↔ (𝐴𝑀) = 𝑀)
6058, 59sylib 220 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝐴𝑀) = 𝑀)
6142, 54, 603eqtr3rd 2863 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 1081   = wceq 1530  wcel 2107  wne 3014  cin 3933  wss 3934  c0 4289  {csn 4559  cfv 6348  (class class class)co 7148  lecple 16564  joincjn 17546  Atomscatm 36391  HLchlt 36478  +𝑃cpadd 36923  𝑃cpolN 37030  PSubClcpscN 37062
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-riotaBAD 36081
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-undef 7931  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-oposet 36304  df-ol 36306  df-oml 36307  df-covers 36394  df-ats 36395  df-atl 36426  df-cvlat 36450  df-hlat 36479  df-psubsp 36631  df-pmap 36632  df-padd 36924  df-polarityN 37031  df-psubclN 37063
This theorem is referenced by:  osumcllem11N  37094
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