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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 483 | . . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → 𝐾 ∈ 𝐵) | |
2 | 0ss 4354 | . . . . 5 ⊢ ∅ ⊆ 𝐴 | |
3 | 2 | a1i 11 | . . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → ∅ ⊆ 𝐴) |
4 | simpr 485 | . . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ 𝐴) | |
5 | 1, 3, 4 | 3jca 1128 | . . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝐾 ∈ 𝐵 ∧ ∅ ⊆ 𝐴 ∧ 𝑋 ⊆ 𝐴)) |
6 | neirr 2950 | . . . 4 ⊢ ¬ ∅ ≠ ∅ | |
7 | 6 | intnanr 488 | . . 3 ⊢ ¬ (∅ ≠ ∅ ∧ 𝑋 ≠ ∅) |
8 | padd0.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | padd0.p | . . . 4 ⊢ + = (+𝑃‘𝐾) | |
10 | 8, 9 | paddval0 38240 | . . 3 ⊢ (((𝐾 ∈ 𝐵 ∧ ∅ ⊆ 𝐴 ∧ 𝑋 ⊆ 𝐴) ∧ ¬ (∅ ≠ ∅ ∧ 𝑋 ≠ ∅)) → (∅ + 𝑋) = (∅ ∪ 𝑋)) |
11 | 5, 7, 10 | sylancl 586 | . 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (∅ + 𝑋) = (∅ ∪ 𝑋)) |
12 | uncom 4111 | . . 3 ⊢ (∅ ∪ 𝑋) = (𝑋 ∪ ∅) | |
13 | un0 4348 | . . 3 ⊢ (𝑋 ∪ ∅) = 𝑋 | |
14 | 12, 13 | eqtri 2764 | . 2 ⊢ (∅ ∪ 𝑋) = 𝑋 |
15 | 11, 14 | eqtrdi 2792 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (∅ + 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∪ cun 3906 ⊆ wss 3908 ∅c0 4280 ‘cfv 6493 (class class class)co 7353 Atomscatm 37692 +𝑃cpadd 38225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7917 df-2nd 7918 df-padd 38226 |
This theorem is referenced by: paddasslem17 38266 pmodlem2 38277 pmapjat1 38283 osumclN 38397 pexmidALTN 38408 |
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