Theorem mptelee 28822

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 28821	. 2 ⊢ (𝑁 ∈ ℕ → ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ∈ (𝔼‘𝑁) ↔ (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)):(1...𝑁)⟶ℝ))
2		ovex 7420	. . . . 5 ⊢ (𝐴𝐹𝐵) ∈ V
3		eqid 2729	. . . . 5 ⊢ (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) = (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵))
4	2, 3	fnmpti 6661	. . . 4 ⊢ (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) Fn (1...𝑁)
5		df-f 6515	. . . 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 709	. . 3 ⊢ ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)):(1...𝑁)⟶ℝ ↔ ran (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ⊆ ℝ)
7	3	rnmpt 5921	. . . . 5 ⊢ ran (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) = {𝑎 ∣ ∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)}
8	7	sseq1i 3975	. . . 4 ⊢ (ran (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ⊆ ℝ ↔ {𝑎 ∣ ∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)} ⊆ ℝ)
9		abss 4026	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)} ⊆ ℝ ↔ ∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
10		nfre1 3262	. . . . . . . . 9 ⊢ Ⅎ𝑘∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)
11		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑎 ∈ ℝ
12	10, 11	nfim 1896	. . . . . . . 8 ⊢ Ⅎ𝑘(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ)
13	12	nfal 2322	. . . . . . 7 ⊢ Ⅎ𝑘∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ)
14		r19.23v 3161	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (1...𝑁)(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) ↔ (∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
15	14	albii 1819	. . . . . . . 8 ⊢ (∀𝑎∀𝑘 ∈ (1...𝑁)(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) ↔ ∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
16		ralcom4 3263	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (1...𝑁)∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) ↔ ∀𝑎∀𝑘 ∈ (1...𝑁)(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
17		rsp 3225	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ (1...𝑁)∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → (𝑘 ∈ (1...𝑁) → ∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ)))
18	2	clel2 3626	. . . . . . . . . 10 ⊢ ((𝐴𝐹𝐵) ∈ ℝ ↔ ∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
19	17, 18	imbitrrdi 252	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (1...𝑁)∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → (𝑘 ∈ (1...𝑁) → (𝐴𝐹𝐵) ∈ ℝ))
20	16, 19	sylbir 235	. . . . . . . 8 ⊢ (∀𝑎∀𝑘 ∈ (1...𝑁)(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → (𝑘 ∈ (1...𝑁) → (𝐴𝐹𝐵) ∈ ℝ))
21	15, 20	sylbir 235	. . . . . . 7 ⊢ (∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → (𝑘 ∈ (1...𝑁) → (𝐴𝐹𝐵) ∈ ℝ))
22	13, 21	ralrimi 3235	. . . . . 6 ⊢ (∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ)
23		nfra1 3261	. . . . . . . 8 ⊢ Ⅎ𝑘∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ
24		rsp 3225	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ → (𝑘 ∈ (1...𝑁) → (𝐴𝐹𝐵) ∈ ℝ))
25		eleq1a 2823	. . . . . . . . 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 3244	. . . . . . 7 ⊢ (∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ → (∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
28	27	alrimiv 1927	. . . . . 6 ⊢ (∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ → ∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
29	22, 28	impbii 209	. . . . 5 ⊢ (∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ)
30	9, 29	bitri 275	. . . 4 ⊢ ({𝑎 ∣ ∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)} ⊆ ℝ ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ)
31	8, 30	bitri 275	. . 3 ⊢ (ran (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ⊆ ℝ ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ)
32	6, 31	bitri 275	. 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 206  ∀wal 1538   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053   ⊆ wss 3914   ↦ cmpt 5188  ran crn 5639   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  ℝcr 11067  1c1 11069  ℕcn 12186  ...cfz 13468  𝔼cee 28815

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-map 8801  df-ee 28818

This theorem is referenced by:  eleesub  28838  eleesubd  28839  axsegconlem1  28844  axsegconlem8  28851  axpasch  28868  axeuclidlem  28889  axcontlem2  28892