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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 4347 | . . . . 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 1120 | . . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝐾 ∈ 𝐵 ∧ ∅ ⊆ 𝐴 ∧ 𝑋 ⊆ 𝐴)) |
6 | neirr 3022 | . . . 4 ⊢ ¬ ∅ ≠ ∅ | |
7 | 6 | intnanr 488 | . . 3 ⊢ ¬ (∅ ≠ ∅ ∧ 𝑋 ≠ ∅) |
8 | padd0.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | padd0.p | . . . 4 ⊢ + = (+𝑃‘𝐾) | |
10 | 8, 9 | paddval0 36826 | . . 3 ⊢ (((𝐾 ∈ 𝐵 ∧ ∅ ⊆ 𝐴 ∧ 𝑋 ⊆ 𝐴) ∧ ¬ (∅ ≠ ∅ ∧ 𝑋 ≠ ∅)) → (∅ + 𝑋) = (∅ ∪ 𝑋)) |
11 | 5, 7, 10 | sylancl 586 | . 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (∅ + 𝑋) = (∅ ∪ 𝑋)) |
12 | uncom 4126 | . . 3 ⊢ (∅ ∪ 𝑋) = (𝑋 ∪ ∅) | |
13 | un0 4341 | . . 3 ⊢ (𝑋 ∪ ∅) = 𝑋 | |
14 | 12, 13 | eqtri 2841 | . 2 ⊢ (∅ ∪ 𝑋) = 𝑋 |
15 | 11, 14 | syl6eq 2869 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (∅ + 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∪ cun 3931 ⊆ wss 3933 ∅c0 4288 ‘cfv 6348 (class class class)co 7145 Atomscatm 36279 +𝑃cpadd 36811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-padd 36812 |
This theorem is referenced by: paddasslem17 36852 pmodlem2 36863 pmapjat1 36869 osumclN 36983 pexmidALTN 36994 |
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