paddssw1 - Mathbox for Norm Megill

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Theorem paddssw1 40042

Description: Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)

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

**Assertion**
Ref	Expression
paddssw1	⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑋 ⊆ 𝑍 ∧ 𝑌 ⊆ 𝑍) → (𝑋 + 𝑌) ⊆ (𝑍 + 𝑍)))

**Proof of Theorem paddssw1**
Step	Hyp	Ref	Expression
1		simpl 482	. 2 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝐾 ∈ 𝐵)
2		simpr3 1197	. 2 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝑍 ⊆ 𝐴)
3		paddssw.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
4		paddssw.p	. . 3 ⊢ + = (+_𝑃‘𝐾)
5	3, 4	paddss12 40018	. 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑍 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → ((𝑋 ⊆ 𝑍 ∧ 𝑌 ⊆ 𝑍) → (𝑋 + 𝑌) ⊆ (𝑍 + 𝑍)))
6	1, 2, 2, 5	syl3anc 1373	1 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑋 ⊆ 𝑍 ∧ 𝑌 ⊆ 𝑍) → (𝑋 + 𝑌) ⊆ (𝑍 + 𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ⊆ wss 3899 ‘cfv 6490 (class class class)co 7356 Atomscatm 39462 +_𝑃cpadd 39994

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7931 df-2nd 7932 df-padd 39995

This theorem is referenced by: paddss 40044