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Mirrors > Home > MPE Home > Th. List > axlowdimlem5 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 26191. 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 26175 | . . . . 5 ⊢ {〈1, 𝐴〉, 〈2, 𝐵〉}:(1...2)⟶ℝ |
4 | axlowdimlem1 26172 | . . . . 5 ⊢ ((3...𝑁) × {0}):(3...𝑁)⟶ℝ | |
5 | 3, 4 | pm3.2i 463 | . . . 4 ⊢ ({〈1, 𝐴〉, 〈2, 𝐵〉}:(1...2)⟶ℝ ∧ ((3...𝑁) × {0}):(3...𝑁)⟶ℝ) |
6 | axlowdimlem2 26173 | . . . 4 ⊢ ((1...2) ∩ (3...𝑁)) = ∅ | |
7 | fun2 6281 | . . . 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 684 | . . 3 ⊢ ({〈1, 𝐴〉, 〈2, 𝐵〉} ∪ ((3...𝑁) × {0})):((1...2) ∪ (3...𝑁))⟶ℝ |
9 | axlowdimlem3 26174 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘2) → (1...𝑁) = ((1...2) ∪ (3...𝑁))) | |
10 | 9 | feq2d 6241 | . . 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 250 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘2) → ({〈1, 𝐴〉, 〈2, 𝐵〉} ∪ ((3...𝑁) × {0})):(1...𝑁)⟶ℝ) |
12 | eluz2nn 11967 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ) | |
13 | elee 26124 | . . 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 249 | 1 ⊢ (𝑁 ∈ (ℤ≥‘2) → ({〈1, 𝐴〉, 〈2, 𝐵〉} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∪ cun 3766 ∩ cin 3767 ∅c0 4114 {csn 4367 {cpr 4369 〈cop 4373 × cxp 5309 ⟶wf 6096 ‘cfv 6100 (class class class)co 6877 ℝcr 10222 0cc0 10223 1c1 10224 ℕcn 11311 2c2 11365 3c3 11366 ℤ≥cuz 11927 ...cfz 12577 𝔼cee 26118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-cnex 10279 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 ax-pre-mulgt0 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-pss 3784 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-tp 4372 df-op 4374 df-uni 4628 df-iun 4711 df-br 4843 df-opab 4905 df-mpt 4922 df-tr 4945 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5897 df-ord 5943 df-on 5944 df-lim 5945 df-suc 5946 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-riota 6838 df-ov 6880 df-oprab 6881 df-mpt2 6882 df-om 7299 df-1st 7400 df-2nd 7401 df-wrecs 7644 df-recs 7706 df-rdg 7744 df-er 7981 df-map 8096 df-en 8195 df-dom 8196 df-sdom 8197 df-pnf 10364 df-mnf 10365 df-xr 10366 df-ltxr 10367 df-le 10368 df-sub 10557 df-neg 10558 df-nn 11312 df-2 11373 df-3 11374 df-n0 11578 df-z 11664 df-uz 11928 df-fz 12578 df-ee 26121 |
This theorem is referenced by: axlowdimlem6 26177 axlowdimlem17 26188 axlowdim2 26190 axlowdim 26191 |
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