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Mirrors > Home > MPE Home > Th. List > Mathboxes > paddss | Structured version Visualization version GIF version |
Description: Subset law for projective subspace sum. (unss 4162 analog.) (Contributed by NM, 7-Mar-2012.) |
Ref | Expression |
---|---|
paddss.a | ⊢ 𝐴 = (Atoms‘𝐾) |
paddss.s | ⊢ 𝑆 = (PSubSp‘𝐾) |
paddss.p | ⊢ + = (+𝑃‘𝐾) |
Ref | Expression |
---|---|
paddss | ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆)) → ((𝑋 ⊆ 𝑍 ∧ 𝑌 ⊆ 𝑍) ↔ (𝑋 + 𝑌) ⊆ 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆)) → 𝐾 ∈ 𝐵) | |
2 | simpr1 1190 | . . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆)) → 𝑋 ⊆ 𝐴) | |
3 | simpr2 1191 | . . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆)) → 𝑌 ⊆ 𝐴) | |
4 | paddss.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | paddss.s | . . . . . 6 ⊢ 𝑆 = (PSubSp‘𝐾) | |
6 | 4, 5 | psubssat 36892 | . . . . 5 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑍 ∈ 𝑆) → 𝑍 ⊆ 𝐴) |
7 | 6 | 3ad2antr3 1186 | . . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆)) → 𝑍 ⊆ 𝐴) |
8 | paddss.p | . . . . 5 ⊢ + = (+𝑃‘𝐾) | |
9 | 4, 8 | paddssw1 36981 | . . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑋 ⊆ 𝑍 ∧ 𝑌 ⊆ 𝑍) → (𝑋 + 𝑌) ⊆ (𝑍 + 𝑍))) |
10 | 1, 2, 3, 7, 9 | syl13anc 1368 | . . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆)) → ((𝑋 ⊆ 𝑍 ∧ 𝑌 ⊆ 𝑍) → (𝑋 + 𝑌) ⊆ (𝑍 + 𝑍))) |
11 | 5, 8 | paddidm 36979 | . . . . 5 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑍 ∈ 𝑆) → (𝑍 + 𝑍) = 𝑍) |
12 | 11 | 3ad2antr3 1186 | . . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆)) → (𝑍 + 𝑍) = 𝑍) |
13 | 12 | sseq2d 4001 | . . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆)) → ((𝑋 + 𝑌) ⊆ (𝑍 + 𝑍) ↔ (𝑋 + 𝑌) ⊆ 𝑍)) |
14 | 10, 13 | sylibd 241 | . 2 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆)) → ((𝑋 ⊆ 𝑍 ∧ 𝑌 ⊆ 𝑍) → (𝑋 + 𝑌) ⊆ 𝑍)) |
15 | 4, 8 | paddssw2 36982 | . . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑋 + 𝑌) ⊆ 𝑍 → (𝑋 ⊆ 𝑍 ∧ 𝑌 ⊆ 𝑍))) |
16 | 1, 2, 3, 7, 15 | syl13anc 1368 | . 2 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆)) → ((𝑋 + 𝑌) ⊆ 𝑍 → (𝑋 ⊆ 𝑍 ∧ 𝑌 ⊆ 𝑍))) |
17 | 14, 16 | impbid 214 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆)) → ((𝑋 ⊆ 𝑍 ∧ 𝑌 ⊆ 𝑍) ↔ (𝑋 + 𝑌) ⊆ 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 ‘cfv 6357 (class class class)co 7158 Atomscatm 36401 PSubSpcpsubsp 36634 +𝑃cpadd 36933 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-psubsp 36641 df-padd 36934 |
This theorem is referenced by: pmodlem1 36984 pclunN 37036 osumcllem1N 37094 |
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