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Mirrors > Home > MPE Home > Th. List > axlowdimlem7 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 26674. 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 2818 | . . . . . . . 8 ⊢ {〈3, -1〉} = {〈3, -1〉} | |
3 | 3ex 11707 | . . . . . . . . 9 ⊢ 3 ∈ V | |
4 | negex 10872 | . . . . . . . . 9 ⊢ -1 ∈ V | |
5 | 3, 4 | fsn 6889 | . . . . . . . 8 ⊢ ({〈3, -1〉}:{3}⟶{-1} ↔ {〈3, -1〉} = {〈3, -1〉}) |
6 | 2, 5 | mpbir 232 | . . . . . . 7 ⊢ {〈3, -1〉}:{3}⟶{-1} |
7 | neg1rr 11740 | . . . . . . . 8 ⊢ -1 ∈ ℝ | |
8 | snssi 4733 | . . . . . . . 8 ⊢ (-1 ∈ ℝ → {-1} ⊆ ℝ) | |
9 | 7, 8 | ax-mp 5 | . . . . . . 7 ⊢ {-1} ⊆ ℝ |
10 | fss 6520 | . . . . . . 7 ⊢ (({〈3, -1〉}:{3}⟶{-1} ∧ {-1} ⊆ ℝ) → {〈3, -1〉}:{3}⟶ℝ) | |
11 | 6, 9, 10 | mp2an 688 | . . . . . 6 ⊢ {〈3, -1〉}:{3}⟶ℝ |
12 | 0re 10631 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
13 | 12 | fconst6 6562 | . . . . . 6 ⊢ (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶ℝ |
14 | 11, 13 | pm3.2i 471 | . . . . 5 ⊢ ({〈3, -1〉}:{3}⟶ℝ ∧ (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶ℝ) |
15 | disjdif 4417 | . . . . 5 ⊢ ({3} ∩ ((1...𝑁) ∖ {3})) = ∅ | |
16 | fun2 6534 | . . . . 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 688 | . . . 4 ⊢ ({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0})):({3} ∪ ((1...𝑁) ∖ {3}))⟶ℝ |
18 | eluzle 12244 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘3) → 3 ≤ 𝑁) | |
19 | 1le3 11837 | . . . . . . . . 9 ⊢ 1 ≤ 3 | |
20 | 18, 19 | jctil 520 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘3) → (1 ≤ 3 ∧ 3 ≤ 𝑁)) |
21 | 3z 12003 | . . . . . . . . 9 ⊢ 3 ∈ ℤ | |
22 | 1z 12000 | . . . . . . . . 9 ⊢ 1 ∈ ℤ | |
23 | eluzelz 12241 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℤ) | |
24 | elfz 12886 | . . . . . . . . 9 ⊢ ((3 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (3 ∈ (1...𝑁) ↔ (1 ≤ 3 ∧ 3 ≤ 𝑁))) | |
25 | 21, 22, 23, 24 | mp3an12i 1456 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘3) → (3 ∈ (1...𝑁) ↔ (1 ≤ 3 ∧ 3 ≤ 𝑁))) |
26 | 20, 25 | mpbird 258 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘3) → 3 ∈ (1...𝑁)) |
27 | 26 | snssd 4734 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘3) → {3} ⊆ (1...𝑁)) |
28 | undif 4426 | . . . . . 6 ⊢ ({3} ⊆ (1...𝑁) ↔ ({3} ∪ ((1...𝑁) ∖ {3})) = (1...𝑁)) | |
29 | 27, 28 | sylib 219 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘3) → ({3} ∪ ((1...𝑁) ∖ {3})) = (1...𝑁)) |
30 | 29 | feq2d 6493 | . . . 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 234 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘3) → ({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0})):(1...𝑁)⟶ℝ) |
32 | eluzge3nn 12278 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ) | |
33 | elee 26607 | . . . 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 258 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘3) → ({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0})) ∈ (𝔼‘𝑁)) |
36 | 1, 35 | eqeltrid 2914 | 1 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑃 ∈ (𝔼‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∖ cdif 3930 ∪ cun 3931 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 {csn 4557 〈cop 4563 class class class wbr 5057 × cxp 5546 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 0cc0 10525 1c1 10526 ≤ cle 10664 -cneg 10859 ℕcn 11626 3c3 11681 ℤcz 11969 ℤ≥cuz 12231 ...cfz 12880 𝔼cee 26601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-z 11970 df-uz 12232 df-fz 12881 df-ee 26604 |
This theorem is referenced by: axlowdimlem15 26669 axlowdimlem16 26670 axlowdimlem17 26671 |
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