osumcllem10N - Mathbox for Norm Megill

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Theorem osumcllem10N 39924

Description: Lemma for osumclN 39926. Contradict osumcllem9N 39923. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
osumcllem.l	⊢ ≤ = (le‘𝐾)
osumcllem.j	⊢ ∨ = (join‘𝐾)
osumcllem.a	⊢ 𝐴 = (Atoms‘𝐾)
osumcllem.p	⊢ + = (+_𝑃‘𝐾)
osumcllem.o	⊢ ⊥ = (⊥_𝑃‘𝐾)
osumcllem.c	⊢ 𝐶 = (PSubCl‘𝐾)
osumcllem.m	⊢ 𝑀 = (𝑋 + {𝑝})
osumcllem.u	⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))

**Assertion**
Ref	Expression
osumcllem10N	⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀 ≠ 𝑋)

**Proof of Theorem osumcllem10N**
Step	Hyp	Ref	Expression
1		simp11 1203	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝐾 ∈ HL)
2		simp2 1137	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑝 ∈ 𝐴)
3	2	snssd 4834	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → {𝑝} ⊆ 𝐴)
4		simp12 1204	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋 ⊆ 𝐴)
5		osumcllem.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
6		osumcllem.p	. . . . . 6 ⊢ + = (+_𝑃‘𝐾)
7	5, 6	sspadd2 39775	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ {𝑝} ⊆ 𝐴 ∧ 𝑋 ⊆ 𝐴) → {𝑝} ⊆ (𝑋 + {𝑝}))
8	1, 3, 4, 7	syl3anc 1371	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → {𝑝} ⊆ (𝑋 + {𝑝}))
9		vex 3492	. . . . 5 ⊢ 𝑝 ∈ V
10	9	snss 4810	. . . 4 ⊢ (𝑝 ∈ (𝑋 + {𝑝}) ↔ {𝑝} ⊆ (𝑋 + {𝑝}))
11	8, 10	sylibr 234	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑝 ∈ (𝑋 + {𝑝}))
12		osumcllem.m	. . 3 ⊢ 𝑀 = (𝑋 + {𝑝})
13	11, 12	eleqtrrdi 2855	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑝 ∈ 𝑀)
14	5, 6	sspadd1 39774	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ (𝑋 + 𝑌))
15	14	3ad2ant1 1133	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑋 ⊆ (𝑋 + 𝑌))
16		simp3 1138	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ¬ 𝑝 ∈ (𝑋 + 𝑌))
17	15, 16	ssneldd 4011	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → ¬ 𝑝 ∈ 𝑋)
18		nelne1 3045	. 2 ⊢ ((𝑝 ∈ 𝑀 ∧ ¬ 𝑝 ∈ 𝑋) → 𝑀 ≠ 𝑋)
19	13, 17, 18	syl2anc 583	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀 ≠ 𝑋)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ⊆ wss 3976 {csn 4648 ‘cfv 6575 (class class class)co 7450 lecple 17320 joincjn 18383 Atomscatm 39221 HLchlt 39308 +_𝑃cpadd 39754 ⊥_𝑃cpolN 39861 PSubClcpscN 39893

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-ov 7453 df-oprab 7454 df-mpo 7455 df-1st 8032 df-2nd 8033 df-padd 39755

This theorem is referenced by: osumcllem11N 39925

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