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Mirrors > Home > MPE Home > Th. List > Mathboxes > paddssw2 | Structured version Visualization version GIF version |
Description: Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.) |
Ref | Expression |
---|---|
paddssw.a | ⊢ 𝐴 = (Atoms‘𝐾) |
paddssw.p | ⊢ + = (+𝑃‘𝐾) |
Ref | Expression |
---|---|
paddssw2 | ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑋 + 𝑌) ⊆ 𝑍 → (𝑋 ⊆ 𝑍 ∧ 𝑌 ⊆ 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | paddssw.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
2 | paddssw.p | . . . . . 6 ⊢ + = (+𝑃‘𝐾) | |
3 | 1, 2 | sspadd1 36966 | . . . . 5 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ (𝑋 + 𝑌)) |
4 | 3 | 3adant3r3 1180 | . . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝑋 ⊆ (𝑋 + 𝑌)) |
5 | sstr 3975 | . . . 4 ⊢ ((𝑋 ⊆ (𝑋 + 𝑌) ∧ (𝑋 + 𝑌) ⊆ 𝑍) → 𝑋 ⊆ 𝑍) | |
6 | 4, 5 | sylan 582 | . . 3 ⊢ (((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) ∧ (𝑋 + 𝑌) ⊆ 𝑍) → 𝑋 ⊆ 𝑍) |
7 | 6 | ex 415 | . 2 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑋 + 𝑌) ⊆ 𝑍 → 𝑋 ⊆ 𝑍)) |
8 | simpl 485 | . . . . 5 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝐾 ∈ 𝐵) | |
9 | simpr2 1191 | . . . . 5 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝑌 ⊆ 𝐴) | |
10 | simpr1 1190 | . . . . 5 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝑋 ⊆ 𝐴) | |
11 | 1, 2 | sspadd2 36967 | . . . . 5 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ 𝐴) → 𝑌 ⊆ (𝑋 + 𝑌)) |
12 | 8, 9, 10, 11 | syl3anc 1367 | . . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝑌 ⊆ (𝑋 + 𝑌)) |
13 | sstr 3975 | . . . 4 ⊢ ((𝑌 ⊆ (𝑋 + 𝑌) ∧ (𝑋 + 𝑌) ⊆ 𝑍) → 𝑌 ⊆ 𝑍) | |
14 | 12, 13 | sylan 582 | . . 3 ⊢ (((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) ∧ (𝑋 + 𝑌) ⊆ 𝑍) → 𝑌 ⊆ 𝑍) |
15 | 14 | ex 415 | . 2 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑋 + 𝑌) ⊆ 𝑍 → 𝑌 ⊆ 𝑍)) |
16 | 7, 15 | jcad 515 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑋 + 𝑌) ⊆ 𝑍 → (𝑋 ⊆ 𝑍 ∧ 𝑌 ⊆ 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ‘cfv 6355 (class class class)co 7156 Atomscatm 36414 +𝑃cpadd 36946 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-padd 36947 |
This theorem is referenced by: paddss 36996 |
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