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| Description: Lemma for axlowdim 28976. 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 2737 | . . . . . . . 8 ⊢ {〈3, -1〉} = {〈3, -1〉} | |
| 3 | 3ex 12348 | . . . . . . . . 9 ⊢ 3 ∈ V | |
| 4 | negex 11506 | . . . . . . . . 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 12381 | . . . . . . . 8 ⊢ -1 ∈ ℝ | |
| 8 | snssi 4808 | . . . . . . . 8 ⊢ (-1 ∈ ℝ → {-1} ⊆ ℝ) | |
| 9 | 7, 8 | ax-mp 5 | . . . . . . 7 ⊢ {-1} ⊆ ℝ | 
| 10 | fss 6752 | . . . . . . 7 ⊢ (({〈3, -1〉}:{3}⟶{-1} ∧ {-1} ⊆ ℝ) → {〈3, -1〉}:{3}⟶ℝ) | |
| 11 | 6, 9, 10 | mp2an 692 | . . . . . 6 ⊢ {〈3, -1〉}:{3}⟶ℝ | 
| 12 | 0re 11263 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
| 13 | 12 | fconst6 6798 | . . . . . 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 4472 | . . . . 5 ⊢ ({3} ∩ ((1...𝑁) ∖ {3})) = ∅ | |
| 16 | fun2 6771 | . . . . 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 12891 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘3) → 3 ≤ 𝑁) | |
| 19 | 1le3 12478 | . . . . . . . . 9 ⊢ 1 ≤ 3 | |
| 20 | 18, 19 | jctil 519 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘3) → (1 ≤ 3 ∧ 3 ≤ 𝑁)) | 
| 21 | 3z 12650 | . . . . . . . . 9 ⊢ 3 ∈ ℤ | |
| 22 | 1z 12647 | . . . . . . . . 9 ⊢ 1 ∈ ℤ | |
| 23 | eluzelz 12888 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℤ) | |
| 24 | elfz 13553 | . . . . . . . . 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 257 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘3) → 3 ∈ (1...𝑁)) | 
| 27 | 26 | snssd 4809 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘3) → {3} ⊆ (1...𝑁)) | 
| 28 | undif 4482 | . . . . . 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 6722 | . . . 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 12932 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ) | |
| 33 | elee 28909 | . . . 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 2845 | 1 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑃 ∈ (𝔼‘𝑁)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {csn 4626 〈cop 4632 class class class wbr 5143 × cxp 5683 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 ≤ cle 11296 -cneg 11493 ℕcn 12266 3c3 12322 ℤcz 12613 ℤ≥cuz 12878 ...cfz 13547 𝔼cee 28903 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-z 12614 df-uz 12879 df-fz 13548 df-ee 28906 | 
| This theorem is referenced by: axlowdimlem15 28971 axlowdimlem16 28972 axlowdimlem17 28973 | 
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