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Mirrors > Home > MPE Home > Th. List > axlowdimlem7 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 28991. 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 2735 | . . . . . . . 8 ⊢ {〈3, -1〉} = {〈3, -1〉} | |
3 | 3ex 12346 | . . . . . . . . 9 ⊢ 3 ∈ V | |
4 | negex 11504 | . . . . . . . . 9 ⊢ -1 ∈ V | |
5 | 3, 4 | fsn 7155 | . . . . . . . 8 ⊢ ({〈3, -1〉}:{3}⟶{-1} ↔ {〈3, -1〉} = {〈3, -1〉}) |
6 | 2, 5 | mpbir 231 | . . . . . . 7 ⊢ {〈3, -1〉}:{3}⟶{-1} |
7 | neg1rr 12379 | . . . . . . . 8 ⊢ -1 ∈ ℝ | |
8 | snssi 4813 | . . . . . . . 8 ⊢ (-1 ∈ ℝ → {-1} ⊆ ℝ) | |
9 | 7, 8 | ax-mp 5 | . . . . . . 7 ⊢ {-1} ⊆ ℝ |
10 | fss 6753 | . . . . . . 7 ⊢ (({〈3, -1〉}:{3}⟶{-1} ∧ {-1} ⊆ ℝ) → {〈3, -1〉}:{3}⟶ℝ) | |
11 | 6, 9, 10 | mp2an 692 | . . . . . 6 ⊢ {〈3, -1〉}:{3}⟶ℝ |
12 | 0re 11261 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
13 | 12 | fconst6 6799 | . . . . . 6 ⊢ (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶ℝ |
14 | 11, 13 | pm3.2i 470 | . . . . 5 ⊢ ({〈3, -1〉}:{3}⟶ℝ ∧ (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶ℝ) |
15 | disjdif 4478 | . . . . 5 ⊢ ({3} ∩ ((1...𝑁) ∖ {3})) = ∅ | |
16 | fun2 6772 | . . . . 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 12889 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘3) → 3 ≤ 𝑁) | |
19 | 1le3 12476 | . . . . . . . . 9 ⊢ 1 ≤ 3 | |
20 | 18, 19 | jctil 519 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘3) → (1 ≤ 3 ∧ 3 ≤ 𝑁)) |
21 | 3z 12648 | . . . . . . . . 9 ⊢ 3 ∈ ℤ | |
22 | 1z 12645 | . . . . . . . . 9 ⊢ 1 ∈ ℤ | |
23 | eluzelz 12886 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℤ) | |
24 | elfz 13550 | . . . . . . . . 9 ⊢ ((3 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (3 ∈ (1...𝑁) ↔ (1 ≤ 3 ∧ 3 ≤ 𝑁))) | |
25 | 21, 22, 23, 24 | mp3an12i 1464 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘3) → (3 ∈ (1...𝑁) ↔ (1 ≤ 3 ∧ 3 ≤ 𝑁))) |
26 | 20, 25 | mpbird 257 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘3) → 3 ∈ (1...𝑁)) |
27 | 26 | snssd 4814 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘3) → {3} ⊆ (1...𝑁)) |
28 | undif 4488 | . . . . . 6 ⊢ ({3} ⊆ (1...𝑁) ↔ ({3} ∪ ((1...𝑁) ∖ {3})) = (1...𝑁)) | |
29 | 27, 28 | sylib 218 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘3) → ({3} ∪ ((1...𝑁) ∖ {3})) = (1...𝑁)) |
30 | 29 | feq2d 6723 | . . . 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 233 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘3) → ({〈3, -1〉} ∪ (((1...𝑁) ∖ {3}) × {0})):(1...𝑁)⟶ℝ) |
32 | eluzge3nn 12930 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ) | |
33 | elee 28924 | . . . 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 257 | . 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∖ cdif 3960 ∪ cun 3961 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 {csn 4631 〈cop 4637 class class class wbr 5148 × cxp 5687 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 ≤ cle 11294 -cneg 11491 ℕcn 12264 3c3 12320 ℤcz 12611 ℤ≥cuz 12876 ...cfz 13544 𝔼cee 28918 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-z 12612 df-uz 12877 df-fz 13545 df-ee 28921 |
This theorem is referenced by: axlowdimlem15 28986 axlowdimlem16 28987 axlowdimlem17 28988 |
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