padd01 - Mathbox for Norm Megill

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Theorem padd01 36946

Description: Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)

**Hypotheses**
Ref	Expression
padd0.a	⊢ 𝐴 = (Atoms‘𝐾)
padd0.p	⊢ + = (+_𝑃‘𝐾)

**Assertion**
Ref	Expression
padd01	⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑋 + ∅) = 𝑋)

**Proof of Theorem padd01**
Step	Hyp	Ref	Expression
1		simpl 485	. . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → 𝐾 ∈ 𝐵)
2		simpr 487	. . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ 𝐴)
3		0ss 4349	. . . . 5 ⊢ ∅ ⊆ 𝐴
4	3	a1i 11	. . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → ∅ ⊆ 𝐴)
5	1, 2, 4	3jca 1124	. . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ ∅ ⊆ 𝐴))
6		neirr 3025	. . . 4 ⊢ ¬ ∅ ≠ ∅
7	6	intnan 489	. . 3 ⊢ ¬ (𝑋 ≠ ∅ ∧ ∅ ≠ ∅)
8		padd0.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
9		padd0.p	. . . 4 ⊢ + = (+_𝑃‘𝐾)
10	8, 9	paddval0 36945	. . 3 ⊢ (((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ ∅ ⊆ 𝐴) ∧ ¬ (𝑋 ≠ ∅ ∧ ∅ ≠ ∅)) → (𝑋 + ∅) = (𝑋 ∪ ∅))
11	5, 7, 10	sylancl 588	. 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑋 + ∅) = (𝑋 ∪ ∅))
12		un0 4343	. 2 ⊢ (𝑋 ∪ ∅) = 𝑋
13	11, 12	syl6eq 2872	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑋 + ∅) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∪ cun 3933 ⊆ wss 3935 ∅c0 4290 ‘cfv 6354 (class class class)co 7155 Atomscatm 36398 +_𝑃cpadd 36930

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-padd 36931

This theorem is referenced by: paddasslem17 36971 pmodlem2 36982

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