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Mirrors > Home > MPE Home > Th. List > axlowdimlem10 | Structured version Visualization version GIF version |
Description: Lemma for axlowdim 26310. 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 6954 | . . . . . . . . 9 ⊢ (𝐼 + 1) ∈ V | |
2 | 1ex 10372 | . . . . . . . . 9 ⊢ 1 ∈ V | |
3 | 1, 2 | f1osn 6430 | . . . . . . . 8 ⊢ {〈(𝐼 + 1), 1〉}:{(𝐼 + 1)}–1-1-onto→{1} |
4 | f1of 6391 | . . . . . . . 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 10370 | . . . . . . . 8 ⊢ 0 ∈ V | |
7 | 6 | fconst 6341 | . . . . . . 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 6316 | . . . . . 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 6282 | . . . . 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 10376 | . . . . . 6 ⊢ 1 ∈ ℝ | |
16 | snssi 4570 | . . . . . 6 ⊢ (1 ∈ ℝ → {1} ⊆ ℝ) | |
17 | 15, 16 | ax-mp 5 | . . . . 5 ⊢ {1} ⊆ ℝ |
18 | 0re 10378 | . . . . . 6 ⊢ 0 ∈ ℝ | |
19 | snssi 4570 | . . . . . 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 6304 | . . . 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 12712 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐼 ∈ (1...(𝑁 − 1))) → (𝐼 + 1) ∈ (1...𝑁)) | |
25 | 24 | snssd 4571 | . . . . 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 6277 | . . 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 26243 | . . 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 5353 ⟶wf 6131 –1-1-onto→wf1o 6134 ‘cfv 6135 (class class class)co 6922 ℝcr 10271 0cc0 10272 1c1 10273 + caddc 10275 − cmin 10606 ℕcn 11374 ...cfz 12643 𝔼cee 26237 |
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 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
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 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-ee 26240 |
This theorem is referenced by: axlowdimlem14 26304 axlowdimlem15 26305 axlowdimlem16 26306 axlowdimlem17 26307 |
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