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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 486 | . . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → 𝐾 ∈ 𝐵) | |
2 | 0ss 4297 | . . . . 5 ⊢ ∅ ⊆ 𝐴 | |
3 | 2 | a1i 11 | . . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → ∅ ⊆ 𝐴) |
4 | simpr 488 | . . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ 𝐴) | |
5 | 1, 3, 4 | 3jca 1130 | . . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝐾 ∈ 𝐵 ∧ ∅ ⊆ 𝐴 ∧ 𝑋 ⊆ 𝐴)) |
6 | neirr 2941 | . . . 4 ⊢ ¬ ∅ ≠ ∅ | |
7 | 6 | intnanr 491 | . . 3 ⊢ ¬ (∅ ≠ ∅ ∧ 𝑋 ≠ ∅) |
8 | padd0.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | padd0.p | . . . 4 ⊢ + = (+𝑃‘𝐾) | |
10 | 8, 9 | paddval0 37510 | . . 3 ⊢ (((𝐾 ∈ 𝐵 ∧ ∅ ⊆ 𝐴 ∧ 𝑋 ⊆ 𝐴) ∧ ¬ (∅ ≠ ∅ ∧ 𝑋 ≠ ∅)) → (∅ + 𝑋) = (∅ ∪ 𝑋)) |
11 | 5, 7, 10 | sylancl 589 | . 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (∅ + 𝑋) = (∅ ∪ 𝑋)) |
12 | uncom 4053 | . . 3 ⊢ (∅ ∪ 𝑋) = (𝑋 ∪ ∅) | |
13 | un0 4291 | . . 3 ⊢ (𝑋 ∪ ∅) = 𝑋 | |
14 | 12, 13 | eqtri 2759 | . 2 ⊢ (∅ ∪ 𝑋) = 𝑋 |
15 | 11, 14 | eqtrdi 2787 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (∅ + 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∪ cun 3851 ⊆ wss 3853 ∅c0 4223 ‘cfv 6358 (class class class)co 7191 Atomscatm 36963 +𝑃cpadd 37495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-padd 37496 |
This theorem is referenced by: paddasslem17 37536 pmodlem2 37547 pmapjat1 37553 osumclN 37667 pexmidALTN 37678 |
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