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Mirrors > Home > MPE Home > Th. List > axlowdimlem5 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 26062. 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 26046 | . . . . 5 ⊢ {〈1, 𝐴〉, 〈2, 𝐵〉}:(1...2)⟶ℝ |
4 | axlowdimlem1 26043 | . . . . 5 ⊢ ((3...𝑁) × {0}):(3...𝑁)⟶ℝ | |
5 | 3, 4 | pm3.2i 456 | . . . 4 ⊢ ({〈1, 𝐴〉, 〈2, 𝐵〉}:(1...2)⟶ℝ ∧ ((3...𝑁) × {0}):(3...𝑁)⟶ℝ) |
6 | axlowdimlem2 26044 | . . . 4 ⊢ ((1...2) ∩ (3...𝑁)) = ∅ | |
7 | fun2 6208 | . . . 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 672 | . . 3 ⊢ ({〈1, 𝐴〉, 〈2, 𝐵〉} ∪ ((3...𝑁) × {0})):((1...2) ∪ (3...𝑁))⟶ℝ |
9 | axlowdimlem3 26045 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘2) → (1...𝑁) = ((1...2) ∪ (3...𝑁))) | |
10 | 9 | feq2d 6170 | . . 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 248 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘2) → ({〈1, 𝐴〉, 〈2, 𝐵〉} ∪ ((3...𝑁) × {0})):(1...𝑁)⟶ℝ) |
12 | eluz2nn 11933 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ) | |
13 | elee 25995 | . . 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 247 | 1 ⊢ (𝑁 ∈ (ℤ≥‘2) → ({〈1, 𝐴〉, 〈2, 𝐵〉} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∪ cun 3721 ∩ cin 3722 ∅c0 4063 {csn 4317 {cpr 4319 〈cop 4323 × cxp 5248 ⟶wf 6026 ‘cfv 6030 (class class class)co 6796 ℝcr 10141 0cc0 10142 1c1 10143 ℕcn 11226 2c2 11276 3c3 11277 ℤ≥cuz 11893 ...cfz 12533 𝔼cee 25989 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-1st 7319 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-er 7900 df-map 8015 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-nn 11227 df-2 11285 df-3 11286 df-n0 11500 df-z 11585 df-uz 11894 df-fz 12534 df-ee 25992 |
This theorem is referenced by: axlowdimlem6 26048 axlowdimlem17 26059 axlowdim2 26061 axlowdim 26062 |
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