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Mirrors > Home > MPE Home > Th. List > Mathboxes > padd02 | Structured version Visualization version GIF version |
Description: Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.) |
Ref | Expression |
---|---|
padd0.a | ⊢ 𝐴 = (Atoms‘𝐾) |
padd0.p | ⊢ + = (+𝑃‘𝐾) |
Ref | Expression |
---|---|
padd02 | ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (∅ + 𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 474 | . . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → 𝐾 ∈ 𝐵) | |
2 | 0ss 4115 | . . . . 5 ⊢ ∅ ⊆ 𝐴 | |
3 | 2 | a1i 11 | . . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → ∅ ⊆ 𝐴) |
4 | simpr 479 | . . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ 𝐴) | |
5 | 1, 3, 4 | 3jca 1123 | . . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝐾 ∈ 𝐵 ∧ ∅ ⊆ 𝐴 ∧ 𝑋 ⊆ 𝐴)) |
6 | neirr 2941 | . . . 4 ⊢ ¬ ∅ ≠ ∅ | |
7 | 6 | intnanr 999 | . . 3 ⊢ ¬ (∅ ≠ ∅ ∧ 𝑋 ≠ ∅) |
8 | padd0.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | padd0.p | . . . 4 ⊢ + = (+𝑃‘𝐾) | |
10 | 8, 9 | paddval0 35599 | . . 3 ⊢ (((𝐾 ∈ 𝐵 ∧ ∅ ⊆ 𝐴 ∧ 𝑋 ⊆ 𝐴) ∧ ¬ (∅ ≠ ∅ ∧ 𝑋 ≠ ∅)) → (∅ + 𝑋) = (∅ ∪ 𝑋)) |
11 | 5, 7, 10 | sylancl 697 | . 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (∅ + 𝑋) = (∅ ∪ 𝑋)) |
12 | uncom 3900 | . . 3 ⊢ (∅ ∪ 𝑋) = (𝑋 ∪ ∅) | |
13 | un0 4110 | . . 3 ⊢ (𝑋 ∪ ∅) = 𝑋 | |
14 | 12, 13 | eqtri 2782 | . 2 ⊢ (∅ ∪ 𝑋) = 𝑋 |
15 | 11, 14 | syl6eq 2810 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (∅ + 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ∪ cun 3713 ⊆ wss 3715 ∅c0 4058 ‘cfv 6049 (class class class)co 6813 Atomscatm 35053 +𝑃cpadd 35584 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-1st 7333 df-2nd 7334 df-padd 35585 |
This theorem is referenced by: paddasslem17 35625 pmodlem2 35636 pmapjat1 35642 osumclN 35756 pexmidALTN 35767 |
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