Theorem mpteleeOLD 28983

Description: Obsolete version of mptelee 28982 as of 2-Feb-2026. (Contributed by Scott Fenton, 7-Jun-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
mpteleeOLD	⊢ (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 mpteleeOLD

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elee 28981	. 2 ⊢ (𝑁 ∈ ℕ → ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ∈ (𝔼‘𝑁) ↔ (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)):(1...𝑁)⟶ℝ))
2		ovex 7391	. . . . 5 ⊢ (𝐴𝐹𝐵) ∈ V
3		eqid 2737	. . . . 5 ⊢ (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) = (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵))
4	2, 3	fnmpti 6633	. . . 4 ⊢ (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) Fn (1...𝑁)
5		df-f 6494	. . . 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 710	. . 3 ⊢ ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)):(1...𝑁)⟶ℝ ↔ ran (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ⊆ ℝ)
7	3	rnmpt 5904	. . . . 5 ⊢ ran (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) = {𝑎 ∣ ∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)}
8	7	sseq1i 3951	. . . 4 ⊢ (ran (𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ⊆ ℝ ↔ {𝑎 ∣ ∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)} ⊆ ℝ)
9		abss 4003	. . . . 5 ⊢ ({𝑎 ∣ ∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)} ⊆ ℝ ↔ ∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
10		nfre1 3263	. . . . . . . . 9 ⊢ Ⅎ𝑘∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵)
11		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑎 ∈ ℝ
12	10, 11	nfim 1898	. . . . . . . 8 ⊢ Ⅎ𝑘(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ)
13	12	nfal 2329	. . . . . . 7 ⊢ Ⅎ𝑘∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ)
14		r19.23v 3165	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (1...𝑁)(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) ↔ (∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
15	14	albii 1821	. . . . . . . 8 ⊢ (∀𝑎∀𝑘 ∈ (1...𝑁)(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) ↔ ∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
16		ralcom4 3264	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (1...𝑁)∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) ↔ ∀𝑎∀𝑘 ∈ (1...𝑁)(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
17		rsp 3226	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ (1...𝑁)∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → (𝑘 ∈ (1...𝑁) → ∀𝑎(𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ)))
18	2	clel2 3603	. . . . . . . . . 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 3236	. . . . . 6 ⊢ (∀𝑎(∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ) → ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ)
23		nfra1 3262	. . . . . . . 8 ⊢ Ⅎ𝑘∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ
24		rsp 3226	. . . . . . . . 9 ⊢ (∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ → (𝑘 ∈ (1...𝑁) → (𝐴𝐹𝐵) ∈ ℝ))
25		eleq1a 2832	. . . . . . . . 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 3245	. . . . . . 7 ⊢ (∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ → (∃𝑘 ∈ (1...𝑁)𝑎 = (𝐴𝐹𝐵) → 𝑎 ∈ ℝ))
28	27	alrimiv 1929	. . . . . 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 1540   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062   ⊆ wss 3890   ↦ cmpt 5167  ran crn 5623   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490  (class class class)co 7358  ℝcr 11026  1c1 11028  ℕcn 12163  ...cfz 13450  𝔼cee 28975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-cnex 11083  ax-resscn 11084

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8766  df-ee 28978

This theorem is referenced by: (None)