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Mirrors > Home > MPE Home > Th. List > axlowdimlem10 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 26314. Set up a family of points in Euclidean space. (Contributed by Scott Fenton, 21-Apr-2013.) |
Ref | Expression |
---|---|
axlowdimlem10.1 | ⊢ 𝑄 = ({〈(𝐼 + 1), 1〉} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) |
Ref | Expression |
---|---|
axlowdimlem10 | ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝑄 ∈ (𝔼‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 6956 | . . . . . . . . 9 ⊢ (𝐼 + 1) ∈ V | |
2 | 1ex 10374 | . . . . . . . . 9 ⊢ 1 ∈ V | |
3 | 1, 2 | f1osn 6432 | . . . . . . . 8 ⊢ {〈(𝐼 + 1), 1〉}:{(𝐼 + 1)}–1-1-onto→{1} |
4 | f1of 6393 | . . . . . . . 8 ⊢ ({〈(𝐼 + 1), 1〉}:{(𝐼 + 1)}–1-1-onto→{1} → {〈(𝐼 + 1), 1〉}:{(𝐼 + 1)}⟶{1}) | |
5 | 3, 4 | ax-mp 5 | . . . . . . 7 ⊢ {〈(𝐼 + 1), 1〉}:{(𝐼 + 1)}⟶{1} |
6 | c0ex 10372 | . . . . . . . 8 ⊢ 0 ∈ V | |
7 | 6 | fconst 6343 | . . . . . . 7 ⊢ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}):((1...𝑁) ∖ {(𝐼 + 1)})⟶{0} |
8 | 5, 7 | pm3.2i 464 | . . . . . 6 ⊢ ({〈(𝐼 + 1), 1〉}:{(𝐼 + 1)}⟶{1} ∧ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}):((1...𝑁) ∖ {(𝐼 + 1)})⟶{0}) |
9 | disjdif 4264 | . . . . . 6 ⊢ ({(𝐼 + 1)} ∩ ((1...𝑁) ∖ {(𝐼 + 1)})) = ∅ | |
10 | fun 6318 | . . . . . 6 ⊢ ((({〈(𝐼 + 1), 1〉}:{(𝐼 + 1)}⟶{1} ∧ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0}):((1...𝑁) ∖ {(𝐼 + 1)})⟶{0}) ∧ ({(𝐼 + 1)} ∩ ((1...𝑁) ∖ {(𝐼 + 1)})) = ∅) → ({〈(𝐼 + 1), 1〉} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})):({(𝐼 + 1)} ∪ ((1...𝑁) ∖ {(𝐼 + 1)}))⟶({1} ∪ {0})) | |
11 | 8, 9, 10 | mp2an 682 | . . . . 5 ⊢ ({〈(𝐼 + 1), 1〉} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})):({(𝐼 + 1)} ∪ ((1...𝑁) ∖ {(𝐼 + 1)}))⟶({1} ∪ {0}) |
12 | axlowdimlem10.1 | . . . . . 6 ⊢ 𝑄 = ({〈(𝐼 + 1), 1〉} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})) | |
13 | 12 | feq1i 6284 | . . . . 5 ⊢ (𝑄:({(𝐼 + 1)} ∪ ((1...𝑁) ∖ {(𝐼 + 1)}))⟶({1} ∪ {0}) ↔ ({〈(𝐼 + 1), 1〉} ∪ (((1...𝑁) ∖ {(𝐼 + 1)}) × {0})):({(𝐼 + 1)} ∪ ((1...𝑁) ∖ {(𝐼 + 1)}))⟶({1} ∪ {0})) |
14 | 11, 13 | mpbir 223 | . . . 4 ⊢ 𝑄:({(𝐼 + 1)} ∪ ((1...𝑁) ∖ {(𝐼 + 1)}))⟶({1} ∪ {0}) |
15 | 1re 10378 | . . . . . 6 ⊢ 1 ∈ ℝ | |
16 | snssi 4572 | . . . . . 6 ⊢ (1 ∈ ℝ → {1} ⊆ ℝ) | |
17 | 15, 16 | ax-mp 5 | . . . . 5 ⊢ {1} ⊆ ℝ |
18 | 0re 10380 | . . . . . 6 ⊢ 0 ∈ ℝ | |
19 | snssi 4572 | . . . . . 6 ⊢ (0 ∈ ℝ → {0} ⊆ ℝ) | |
20 | 18, 19 | ax-mp 5 | . . . . 5 ⊢ {0} ⊆ ℝ |
21 | 17, 20 | unssi 4011 | . . . 4 ⊢ ({1} ∪ {0}) ⊆ ℝ |
22 | fss 6306 | . . . 4 ⊢ ((𝑄:({(𝐼 + 1)} ∪ ((1...𝑁) ∖ {(𝐼 + 1)}))⟶({1} ∪ {0}) ∧ ({1} ∪ {0}) ⊆ ℝ) → 𝑄:({(𝐼 + 1)} ∪ ((1...𝑁) ∖ {(𝐼 + 1)}))⟶ℝ) | |
23 | 14, 21, 22 | mp2an 682 | . . 3 ⊢ 𝑄:({(𝐼 + 1)} ∪ ((1...𝑁) ∖ {(𝐼 + 1)}))⟶ℝ |
24 | fznatpl1 12716 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → (𝐼 + 1) ∈ (1...𝑁)) | |
25 | 24 | snssd 4573 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → {(𝐼 + 1)} ⊆ (1...𝑁)) |
26 | undif 4273 | . . . . 5 ⊢ ({(𝐼 + 1)} ⊆ (1...𝑁) ↔ ({(𝐼 + 1)} ∪ ((1...𝑁) ∖ {(𝐼 + 1)})) = (1...𝑁)) | |
27 | 25, 26 | sylib 210 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → ({(𝐼 + 1)} ∪ ((1...𝑁) ∖ {(𝐼 + 1)})) = (1...𝑁)) |
28 | 27 | feq2d 6279 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → (𝑄:({(𝐼 + 1)} ∪ ((1...𝑁) ∖ {(𝐼 + 1)}))⟶ℝ ↔ 𝑄:(1...𝑁)⟶ℝ)) |
29 | 23, 28 | mpbii 225 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝑄:(1...𝑁)⟶ℝ) |
30 | elee 26247 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑄 ∈ (𝔼‘𝑁) ↔ 𝑄:(1...𝑁)⟶ℝ)) | |
31 | 30 | adantr 474 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → (𝑄 ∈ (𝔼‘𝑁) ↔ 𝑄:(1...𝑁)⟶ℝ)) |
32 | 29, 31 | mpbird 249 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → 𝑄 ∈ (𝔼‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∖ cdif 3789 ∪ cun 3790 ∩ cin 3791 ⊆ wss 3792 ∅c0 4141 {csn 4398 〈cop 4404 × cxp 5355 ⟶wf 6133 –1-1-onto→wf1o 6136 ‘cfv 6137 (class class class)co 6924 ℝcr 10273 0cc0 10274 1c1 10275 + caddc 10277 − cmin 10608 ℕcn 11378 ...cfz 12647 𝔼cee 26241 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11379 df-n0 11647 df-z 11733 df-uz 11997 df-fz 12648 df-ee 26244 |
This theorem is referenced by: axlowdimlem14 26308 axlowdimlem15 26309 axlowdimlem16 26310 axlowdimlem17 26311 |
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