paddssw1 - Mathbox for Norm Megill

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Theorem paddssw1 40645

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 487	. 2 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝐾 ∈ 𝐵)
2		simpr3 1214	. 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 40621	. 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑍 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → ((𝑋 ⊆ 𝑍 ∧ 𝑌 ⊆ 𝑍) → (𝑋 + 𝑌) ⊆ (𝑍 + 𝑍)))
6	1, 2, 2, 5	syl3anc 1397	1 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑋 ⊆ 𝑍 ∧ 𝑌 ⊆ 𝑍) → (𝑋 + 𝑌) ⊆ (𝑍 + 𝑍)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 400 ∧ w3a 1102 = wceq 1569 ∈ wcel 2142 ⊆ wss 3904 ‘cfv 6536 (class class class)co 7412 Atomscatm 40065 +_𝑃cpadd 40597

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-id 5555 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7984 df-2nd 7985 df-padd 40598

This theorem is used by: paddss 40647

W3C validator