Theorem mptelee 27263

Description: A condition for a mapping to be an element of a Euclidean space. (Contributed by Scott Fenton, 7-Jun-2013.)

Assertion

Ref	Expression
mptelee	⊢ (𝑁 ∈ ℕ → ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ∈ (𝔼‘𝑁) ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ))

Distinct variable group:   𝑘,𝑁

Allowed substitution hints:   𝐴(𝑘)   𝐵(𝑘)   𝐹(𝑘)

Proof of Theorem mptelee

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elee 27262	. 2 ⊢ (𝑁 ∈ ℕ → ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ∈ (𝔼‘𝑁) ↔ (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)):(1...𝑁)⟶ℝ))
2		ovex 7308	. . . . 5 ⊢ (𝐴𝐹𝐵) ∈ V
3		eqid 2738	. . . . 5 ⊢ (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) = (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵))
4	2, 3	fnmpti 6576	. . . 4 ⊢ (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) Fn (1...𝑁)
5		df-f 6437	. . . 4 ⊢ ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)):(1...𝑁)⟶ℝ ↔ ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) Fn (1...𝑁) ∧ ran (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ⊆ ℝ))
6	4, 5	mpbiran 706	. . 3 ⊢ ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)):(1...𝑁)⟶ℝ ↔ ran (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ⊆ ℝ)
7	3	rnmpt 5864	. . . . 5 ⊢ ran (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) = {𝑎 ∣ ∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)}
8	7	sseq1i 3949	. . . 4 ⊢ (ran (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ⊆ ℝ ↔ {𝑎 ∣ ∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)} ⊆ ℝ)
9		abss 3994	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)} ⊆ ℝ ↔ ∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
10		nfre1 3239	. . . . . . . . 9 ⊢ Ⅎ𝑘∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)
11		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑎 ∈ ℝ
12	10, 11	nfim 1899	. . . . . . . 8 ⊢ Ⅎ𝑘(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ)
13	12	nfal 2317	. . . . . . 7 ⊢ Ⅎ𝑘∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ)
14		r19.23v 3208	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (1...𝑁)(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) ↔ (∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
15	14	albii 1822	. . . . . . . 8 ⊢ (∀𝑎∀𝑘 ∈ (1...𝑁)(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) ↔ ∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
16		ralcom4 3164	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (1...𝑁)∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) ↔ ∀𝑎∀𝑘 ∈ (1...𝑁)(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
17		rsp 3131	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ (1...𝑁)∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → (𝑘 ∈ (1...𝑁) → ∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ)))
18	2	clel2 3590	. . . . . . . . . 10 ⊢ ((𝐴𝐹𝐵) ∈ ℝ ↔ ∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
19	17, 18	syl6ibr 251	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (1...𝑁)∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → (𝑘 ∈ (1...𝑁) → (𝐴𝐹𝐵) ∈ ℝ))
20	16, 19	sylbir 234	. . . . . . . 8 ⊢ (∀𝑎∀𝑘 ∈ (1...𝑁)(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → (𝑘 ∈ (1...𝑁) → (𝐴𝐹𝐵) ∈ ℝ))
21	15, 20	sylbir 234	. . . . . . 7 ⊢ (∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → (𝑘 ∈ (1...𝑁) → (𝐴𝐹𝐵) ∈ ℝ))
22	13, 21	ralrimi 3141	. . . . . 6 ⊢ (∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ)
23		nfra1 3144	. . . . . . . 8 ⊢ Ⅎ𝑘∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ
24		rsp 3131	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ → (𝑘 ∈ (1...𝑁) → (𝐴𝐹𝐵) ∈ ℝ))
25		eleq1a 2834	. . . . . . . . 9 ⊢ ((𝐴𝐹𝐵) ∈ ℝ → (𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
26	24, 25	syl6 35	. . . . . . . 8 ⊢ (∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ → (𝑘 ∈ (1...𝑁) → (𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ)))
27	23, 11, 26	rexlimd 3250	. . . . . . 7 ⊢ (∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ → (∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
28	27	alrimiv 1930	. . . . . 6 ⊢ (∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ → ∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
29	22, 28	impbii 208	. . . . 5 ⊢ (∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ)
30	9, 29	bitri 274	. . . 4 ⊢ ({𝑎 ∣ ∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)} ⊆ ℝ ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ)
31	8, 30	bitri 274	. . 3 ⊢ (ran (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ⊆ ℝ ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ)
32	6, 31	bitri 274	. 2 ⊢ ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)):(1...𝑁)⟶ℝ ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ)
33	1, 32	bitrdi 287	1 ⊢ (𝑁 ∈ ℕ → ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ∈ (𝔼‘𝑁) ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887   ↦ cmpt 5157  ran crn 5590   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  ℝcr 10870  1c1 10872  ℕcn 11973  ...cfz 13239  𝔼cee 27256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-ee 27259

This theorem is referenced by:  eleesub  27279  eleesubd  27280  axsegconlem1  27285  axsegconlem8  27292  axpasch  27309  axeuclidlem  27330  axcontlem2  27333