Theorem mptelee 26683

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 26682	. 2 ⊢ (𝑁 ∈ ℕ → ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ∈ (𝔼‘𝑁) ↔ (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)):(1...𝑁)⟶ℝ))
2		ovex 7191	. . . . 5 ⊢ (𝐴𝐹𝐵) ∈ V
3		eqid 2823	. . . . 5 ⊢ (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) = (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵))
4	2, 3	fnmpti 6493	. . . 4 ⊢ (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) Fn (1...𝑁)
5		df-f 6361	. . . 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 707	. . 3 ⊢ ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)):(1...𝑁)⟶ℝ ↔ ran (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ⊆ ℝ)
7	3	rnmpt 5829	. . . . 5 ⊢ ran (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) = {𝑎 ∣ ∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)}
8	7	sseq1i 3997	. . . 4 ⊢ (ran (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ⊆ ℝ ↔ {𝑎 ∣ ∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)} ⊆ ℝ)
9		abss 4042	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)} ⊆ ℝ ↔ ∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
10		nfre1 3308	. . . . . . . . 9 ⊢ Ⅎ𝑘∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)
11		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑎 ∈ ℝ
12	10, 11	nfim 1897	. . . . . . . 8 ⊢ Ⅎ𝑘(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ)
13	12	nfal 2342	. . . . . . 7 ⊢ Ⅎ𝑘∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ)
14		r19.23v 3281	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (1...𝑁)(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) ↔ (∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
15	14	albii 1820	. . . . . . . 8 ⊢ (∀𝑎∀𝑘 ∈ (1...𝑁)(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) ↔ ∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
16		ralcom4 3237	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (1...𝑁)∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) ↔ ∀𝑎∀𝑘 ∈ (1...𝑁)(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
17		rsp 3207	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ (1...𝑁)∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → (𝑘 ∈ (1...𝑁) → ∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ)))
18	2	clel2 3655	. . . . . . . . . 10 ⊢ ((𝐴𝐹𝐵) ∈ ℝ ↔ ∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
19	17, 18	syl6ibr 254	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (1...𝑁)∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → (𝑘 ∈ (1...𝑁) → (𝐴𝐹𝐵) ∈ ℝ))
20	16, 19	sylbir 237	. . . . . . . 8 ⊢ (∀𝑎∀𝑘 ∈ (1...𝑁)(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → (𝑘 ∈ (1...𝑁) → (𝐴𝐹𝐵) ∈ ℝ))
21	15, 20	sylbir 237	. . . . . . 7 ⊢ (∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → (𝑘 ∈ (1...𝑁) → (𝐴𝐹𝐵) ∈ ℝ))
22	13, 21	ralrimi 3218	. . . . . 6 ⊢ (∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ)
23		nfra1 3221	. . . . . . . 8 ⊢ Ⅎ𝑘∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ
24		rsp 3207	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ → (𝑘 ∈ (1...𝑁) → (𝐴𝐹𝐵) ∈ ℝ))
25		eleq1a 2910	. . . . . . . . 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 3319	. . . . . . 7 ⊢ (∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ → (∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
28	27	alrimiv 1928	. . . . . 6 ⊢ (∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ → ∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
29	22, 28	impbii 211	. . . . 5 ⊢ (∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ)
30	9, 29	bitri 277	. . . 4 ⊢ ({𝑎 ∣ ∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)} ⊆ ℝ ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ)
31	8, 30	bitri 277	. . 3 ⊢ (ran (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ⊆ ℝ ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ)
32	6, 31	bitri 277	. 2 ⊢ ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)):(1...𝑁)⟶ℝ ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ)
33	1, 32	syl6bb 289	1 ⊢ (𝑁 ∈ ℕ → ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ∈ (𝔼‘𝑁) ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535   = wceq 1537   ∈ wcel 2114  {cab 2801  ∀wral 3140  ∃wrex 3141   ⊆ wss 3938   ↦ cmpt 5148  ran crn 5558   Fn wfn 6352  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  ℝcr 10538  1c1 10540  ℕcn 11640  ...cfz 12895  𝔼cee 26676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-map 8410  df-ee 26679

This theorem is referenced by:  eleesub  26699  eleesubd  26700  axsegconlem1  26705  axsegconlem8  26712  axpasch  26729  axeuclidlem  26750  axcontlem2  26753