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Mirrors > Home > MPE Home > Th. List > axlowdimlem5 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 26755. Show that a particular union is a point in Euclidean space. (Contributed by Scott Fenton, 29-Jun-2013.) |
Ref | Expression |
---|---|
axlowdimlem4.1 | ⊢ 𝐴 ∈ ℝ |
axlowdimlem4.2 | ⊢ 𝐵 ∈ ℝ |
Ref | Expression |
---|---|
axlowdimlem5 | ⊢ (𝑁 ∈ (ℤ≥‘2) → ({〈1, 𝐴〉, 〈2, 𝐵〉} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axlowdimlem4.1 | . . . . . 6 ⊢ 𝐴 ∈ ℝ | |
2 | axlowdimlem4.2 | . . . . . 6 ⊢ 𝐵 ∈ ℝ | |
3 | 1, 2 | axlowdimlem4 26739 | . . . . 5 ⊢ {〈1, 𝐴〉, 〈2, 𝐵〉}:(1...2)⟶ℝ |
4 | axlowdimlem1 26736 | . . . . 5 ⊢ ((3...𝑁) × {0}):(3...𝑁)⟶ℝ | |
5 | 3, 4 | pm3.2i 474 | . . . 4 ⊢ ({〈1, 𝐴〉, 〈2, 𝐵〉}:(1...2)⟶ℝ ∧ ((3...𝑁) × {0}):(3...𝑁)⟶ℝ) |
6 | axlowdimlem2 26737 | . . . 4 ⊢ ((1...2) ∩ (3...𝑁)) = ∅ | |
7 | fun2 6515 | . . . 4 ⊢ ((({〈1, 𝐴〉, 〈2, 𝐵〉}:(1...2)⟶ℝ ∧ ((3...𝑁) × {0}):(3...𝑁)⟶ℝ) ∧ ((1...2) ∩ (3...𝑁)) = ∅) → ({〈1, 𝐴〉, 〈2, 𝐵〉} ∪ ((3...𝑁) × {0})):((1...2) ∪ (3...𝑁))⟶ℝ) | |
8 | 5, 6, 7 | mp2an 691 | . . 3 ⊢ ({〈1, 𝐴〉, 〈2, 𝐵〉} ∪ ((3...𝑁) × {0})):((1...2) ∪ (3...𝑁))⟶ℝ |
9 | axlowdimlem3 26738 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘2) → (1...𝑁) = ((1...2) ∪ (3...𝑁))) | |
10 | 9 | feq2d 6473 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → (({〈1, 𝐴〉, 〈2, 𝐵〉} ∪ ((3...𝑁) × {0})):(1...𝑁)⟶ℝ ↔ ({〈1, 𝐴〉, 〈2, 𝐵〉} ∪ ((3...𝑁) × {0})):((1...2) ∪ (3...𝑁))⟶ℝ)) |
11 | 8, 10 | mpbiri 261 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘2) → ({〈1, 𝐴〉, 〈2, 𝐵〉} ∪ ((3...𝑁) × {0})):(1...𝑁)⟶ℝ) |
12 | eluz2nn 12272 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ) | |
13 | elee 26688 | . . 3 ⊢ (𝑁 ∈ ℕ → (({〈1, 𝐴〉, 〈2, 𝐵〉} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁) ↔ ({〈1, 𝐴〉, 〈2, 𝐵〉} ∪ ((3...𝑁) × {0})):(1...𝑁)⟶ℝ)) | |
14 | 12, 13 | syl 17 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘2) → (({〈1, 𝐴〉, 〈2, 𝐵〉} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁) ↔ ({〈1, 𝐴〉, 〈2, 𝐵〉} ∪ ((3...𝑁) × {0})):(1...𝑁)⟶ℝ)) |
15 | 11, 14 | mpbird 260 | 1 ⊢ (𝑁 ∈ (ℤ≥‘2) → ({〈1, 𝐴〉, 〈2, 𝐵〉} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∪ cun 3879 ∩ cin 3880 ∅c0 4243 {csn 4525 {cpr 4527 〈cop 4531 × cxp 5517 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 ℕcn 11625 2c2 11680 3c3 11681 ℤ≥cuz 12231 ...cfz 12885 𝔼cee 26682 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-ee 26685 |
This theorem is referenced by: axlowdimlem6 26741 axlowdimlem17 26752 axlowdim2 26754 axlowdim 26755 |
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