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Mirrors > Home > MPE Home > Th. List > axlowdimlem7 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 27076. Set up a point in Euclidean space. (Contributed by Scott Fenton, 29-Jun-2013.) |
Ref | Expression |
---|---|
axlowdimlem7.1 | ⊢ 𝑃 = ({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0})) |
Ref | Expression |
---|---|
axlowdimlem7 | ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑃 ∈ (𝔼‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axlowdimlem7.1 | . 2 ⊢ 𝑃 = ({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0})) | |
2 | eqid 2738 | . . . . . . . 8 ⊢ {〈3, -1〉} = {〈3, -1〉} | |
3 | 3ex 11936 | . . . . . . . . 9 ⊢ 3 ∈ V | |
4 | negex 11100 | . . . . . . . . 9 ⊢ -1 ∈ V | |
5 | 3, 4 | fsn 6968 | . . . . . . . 8 ⊢ ({〈3, -1〉}:{3}⟶{-1} ↔ {〈3, -1〉} = {〈3, -1〉}) |
6 | 2, 5 | mpbir 234 | . . . . . . 7 ⊢ {〈3, -1〉}:{3}⟶{-1} |
7 | neg1rr 11969 | . . . . . . . 8 ⊢ -1 ∈ ℝ | |
8 | snssi 4735 | . . . . . . . 8 ⊢ (-1 ∈ ℝ → {-1} ⊆ ℝ) | |
9 | 7, 8 | ax-mp 5 | . . . . . . 7 ⊢ {-1} ⊆ ℝ |
10 | fss 6580 | . . . . . . 7 ⊢ (({〈3, -1〉}:{3}⟶{-1} ∧ {-1} ⊆ ℝ) → {〈3, -1〉}:{3}⟶ℝ) | |
11 | 6, 9, 10 | mp2an 692 | . . . . . 6 ⊢ {〈3, -1〉}:{3}⟶ℝ |
12 | 0re 10859 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
13 | 12 | fconst6 6627 | . . . . . 6 ⊢ (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶ℝ |
14 | 11, 13 | pm3.2i 474 | . . . . 5 ⊢ ({〈3, -1〉}:{3}⟶ℝ ∧ (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶ℝ) |
15 | disjdif 4400 | . . . . 5 ⊢ ({3} ∩ ((1...𝑁) ∖ {3})) = ∅ | |
16 | fun2 6600 | . . . . 5 ⊢ ((({〈3, -1〉}:{3}⟶ℝ ∧ (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶ℝ) ∧ ({3} ∩ ((1...𝑁) ∖ {3})) = ∅) → ({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0})):({3} ∪ ((1...𝑁) ∖ {3}))⟶ℝ) | |
17 | 14, 15, 16 | mp2an 692 | . . . 4 ⊢ ({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0})):({3} ∪ ((1...𝑁) ∖ {3}))⟶ℝ |
18 | eluzle 12475 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘3) → 3 ≤ 𝑁) | |
19 | 1le3 12066 | . . . . . . . . 9 ⊢ 1 ≤ 3 | |
20 | 18, 19 | jctil 523 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘3) → (1 ≤ 3 ∧ 3 ≤ 𝑁)) |
21 | 3z 12234 | . . . . . . . . 9 ⊢ 3 ∈ ℤ | |
22 | 1z 12231 | . . . . . . . . 9 ⊢ 1 ∈ ℤ | |
23 | eluzelz 12472 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℤ) | |
24 | elfz 13125 | . . . . . . . . 9 ⊢ ((3 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (3 ∈ (1...𝑁) ↔ (1 ≤ 3 ∧ 3 ≤ 𝑁))) | |
25 | 21, 22, 23, 24 | mp3an12i 1467 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘3) → (3 ∈ (1...𝑁) ↔ (1 ≤ 3 ∧ 3 ≤ 𝑁))) |
26 | 20, 25 | mpbird 260 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘3) → 3 ∈ (1...𝑁)) |
27 | 26 | snssd 4736 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘3) → {3} ⊆ (1...𝑁)) |
28 | undif 4410 | . . . . . 6 ⊢ ({3} ⊆ (1...𝑁) ↔ ({3} ∪ ((1...𝑁) ∖ {3})) = (1...𝑁)) | |
29 | 27, 28 | sylib 221 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘3) → ({3} ∪ ((1...𝑁) ∖ {3})) = (1...𝑁)) |
30 | 29 | feq2d 6549 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘3) → (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0})):({3} ∪ ((1...𝑁) ∖ {3}))⟶ℝ ↔ ({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0})):(1...𝑁)⟶ℝ)) |
31 | 17, 30 | mpbii 236 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘3) → ({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0})):(1...𝑁)⟶ℝ) |
32 | eluzge3nn 12510 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ) | |
33 | elee 27009 | . . . 4 ⊢ (𝑁 ∈ ℕ → (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0})) ∈ (𝔼‘𝑁) ↔ ({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0})):(1...𝑁)⟶ℝ)) | |
34 | 32, 33 | syl 17 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘3) → (({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0})) ∈ (𝔼‘𝑁) ↔ ({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0})):(1...𝑁)⟶ℝ)) |
35 | 31, 34 | mpbird 260 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘3) → ({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0})) ∈ (𝔼‘𝑁)) |
36 | 1, 35 | eqeltrid 2843 | 1 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑃 ∈ (𝔼‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2111 ∖ cdif 3877 ∪ cun 3878 ∩ cin 3879 ⊆ wss 3880 ∅c0 4251 {csn 4555 〈cop 4561 class class class wbr 5067 × cxp 5563 ⟶wf 6393 ‘cfv 6397 (class class class)co 7231 ℝcr 10752 0cc0 10753 1c1 10754 ≤ cle 10892 -cneg 11087 ℕcn 11854 3c3 11910 ℤcz 12200 ℤ≥cuz 12462 ...cfz 13119 𝔼cee 27003 |
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 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-cnex 10809 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 ax-pre-mulgt0 10830 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-om 7663 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-er 8411 df-map 8530 df-en 8647 df-dom 8648 df-sdom 8649 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-sub 11088 df-neg 11089 df-nn 11855 df-2 11917 df-3 11918 df-z 12201 df-uz 12463 df-fz 13120 df-ee 27006 |
This theorem is referenced by: axlowdimlem15 27071 axlowdimlem16 27072 axlowdimlem17 27073 |
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