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Mirrors > Home > MPE Home > Th. List > axlowdimlem5 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 26741. 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 26725 | . . . . 5 ⊢ {〈1, 𝐴〉, 〈2, 𝐵〉}:(1...2)⟶ℝ |
4 | axlowdimlem1 26722 | . . . . 5 ⊢ ((3...𝑁) × {0}):(3...𝑁)⟶ℝ | |
5 | 3, 4 | pm3.2i 473 | . . . 4 ⊢ ({〈1, 𝐴〉, 〈2, 𝐵〉}:(1...2)⟶ℝ ∧ ((3...𝑁) × {0}):(3...𝑁)⟶ℝ) |
6 | axlowdimlem2 26723 | . . . 4 ⊢ ((1...2) ∩ (3...𝑁)) = ∅ | |
7 | fun2 6535 | . . . 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 690 | . . 3 ⊢ ({〈1, 𝐴〉, 〈2, 𝐵〉} ∪ ((3...𝑁) × {0})):((1...2) ∪ (3...𝑁))⟶ℝ |
9 | axlowdimlem3 26724 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘2) → (1...𝑁) = ((1...2) ∪ (3...𝑁))) | |
10 | 9 | feq2d 6494 | . . 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 260 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘2) → ({〈1, 𝐴〉, 〈2, 𝐵〉} ∪ ((3...𝑁) × {0})):(1...𝑁)⟶ℝ) |
12 | eluz2nn 12278 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ) | |
13 | elee 26674 | . . 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 259 | 1 ⊢ (𝑁 ∈ (ℤ≥‘2) → ({〈1, 𝐴〉, 〈2, 𝐵〉} ∪ ((3...𝑁) × {0})) ∈ (𝔼‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∪ cun 3933 ∩ cin 3934 ∅c0 4290 {csn 4560 {cpr 4562 〈cop 4566 × cxp 5547 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ℝcr 10530 0cc0 10531 1c1 10532 ℕcn 11632 2c2 11686 3c3 11687 ℤ≥cuz 12237 ...cfz 12886 𝔼cee 26668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-ee 26671 |
This theorem is referenced by: axlowdimlem6 26727 axlowdimlem17 26738 axlowdim2 26740 axlowdim 26741 |
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