Theorem mptelee 25575

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 25574	. 2 ⊢ (𝑁 ∈ ℕ → ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ∈ (𝔼‘𝑁) ↔ (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)):(1...𝑁)⟶ℝ))
2		ovex 6577	. . . . 5 ⊢ (𝐴𝐹𝐵) ∈ V
3		eqid 2610	. . . . 5 ⊢ (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) = (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵))
4	2, 3	fnmpti 5935	. . . 4 ⊢ (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) Fn (1...𝑁)
5		df-f 5808	. . . 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 955	. . 3 ⊢ ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)):(1...𝑁)⟶ℝ ↔ ran (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ⊆ ℝ)
7	3	rnmpt 5292	. . . . 5 ⊢ ran (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) = {𝑎 ∣ ∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)}
8	7	sseq1i 3592	. . . 4 ⊢ (ran (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ⊆ ℝ ↔ {𝑎 ∣ ∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)} ⊆ ℝ)
9		abss 3634	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)} ⊆ ℝ ↔ ∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
10		nfre1 2988	. . . . . . . . 9 ⊢ Ⅎ𝑘∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)
11		nfv 1830	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑎 ∈ ℝ
12	10, 11	nfim 1813	. . . . . . . 8 ⊢ Ⅎ𝑘(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ)
13	12	nfal 2139	. . . . . . 7 ⊢ Ⅎ𝑘∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ)
14		r19.23v 3005	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (1...𝑁)(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) ↔ (∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
15	14	albii 1737	. . . . . . . 8 ⊢ (∀𝑎∀𝑘 ∈ (1...𝑁)(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) ↔ ∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
16		ralcom4 3197	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (1...𝑁)∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) ↔ ∀𝑎∀𝑘 ∈ (1...𝑁)(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
17		rsp 2913	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ (1...𝑁)∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → (𝑘 ∈ (1...𝑁) → ∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ)))
18	2	clel2 3309	. . . . . . . . . 10 ⊢ ((𝐴𝐹𝐵) ∈ ℝ ↔ ∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
19	17, 18	syl6ibr 241	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (1...𝑁)∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → (𝑘 ∈ (1...𝑁) → (𝐴𝐹𝐵) ∈ ℝ))
20	16, 19	sylbir 224	. . . . . . . 8 ⊢ (∀𝑎∀𝑘 ∈ (1...𝑁)(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → (𝑘 ∈ (1...𝑁) → (𝐴𝐹𝐵) ∈ ℝ))
21	15, 20	sylbir 224	. . . . . . 7 ⊢ (∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → (𝑘 ∈ (1...𝑁) → (𝐴𝐹𝐵) ∈ ℝ))
22	13, 21	ralrimi 2940	. . . . . 6 ⊢ (∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ)
23		nfra1 2925	. . . . . . . 8 ⊢ Ⅎ𝑘∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ
24		rsp 2913	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ → (𝑘 ∈ (1...𝑁) → (𝐴𝐹𝐵) ∈ ℝ))
25		eleq1a 2683	. . . . . . . . 9 ⊢ ((𝐴𝐹𝐵) ∈ ℝ → (𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
26	24, 25	syl6 34	. . . . . . . 8 ⊢ (∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ → (𝑘 ∈ (1...𝑁) → (𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ)))
27	23, 11, 26	rexlimd 3008	. . . . . . 7 ⊢ (∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ → (∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
28	27	alrimiv 1842	. . . . . 6 ⊢ (∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ → ∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
29	22, 28	impbii 198	. . . . 5 ⊢ (∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ)
30	9, 29	bitri 263	. . . 4 ⊢ ({𝑎 ∣ ∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)} ⊆ ℝ ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ)
31	8, 30	bitri 263	. . 3 ⊢ (ran (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ⊆ ℝ ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ)
32	6, 31	bitri 263	. 2 ⊢ ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)):(1...𝑁)⟶ℝ ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ)
33	1, 32	syl6bb 275	1 ⊢ (𝑁 ∈ ℕ → ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ∈ (𝔼‘𝑁) ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540   ↦ cmpt 4643  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  1c1 9816  ℕcn 10897  ...cfz 12197  𝔼cee 25568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-ee 25571

This theorem is referenced by:  eleesub  25591  eleesubd  25592  axsegconlem1  25597  axsegconlem8  25604  axpasch  25621  axeuclidlem  25642  axcontlem2  25645