Theorem fourierdlem34 46097

Description: A partition is one to one. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fourierdlem34.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem34.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem34.q	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))

Assertion

Ref	Expression
fourierdlem34	⊢ (𝜑 → 𝑄:(0...𝑀)–1-1→ℝ)

Distinct variable groups:   𝐴,𝑚,𝑝   𝐵,𝑚,𝑝   𝑖,𝑀,𝑚,𝑝   𝑄,𝑖,𝑝   𝜑,𝑖

Allowed substitution hints:   𝜑(𝑚,𝑝)   𝐴(𝑖)   𝐵(𝑖)   𝑃(𝑖,𝑚,𝑝)   𝑄(𝑚)

Proof of Theorem fourierdlem34

Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem34.q	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
2		fourierdlem34.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3		fourierdlem34.p	. . . . . . 7 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
4	3	fourierdlem2 46065	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
5	2, 4	syl 17	. . . . 5 ⊢ (𝜑 → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
6	1, 5	mpbid 232	. . . 4 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
7	6	simpld 494	. . 3 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑m (0...𝑀)))
8		elmapi 8888	. . 3 ⊢ (𝑄 ∈ (ℝ ↑m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
9	7, 8	syl 17	. 2 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
10		simplr 769	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = (𝑄‘𝑗)) ∧ ¬ 𝑖 = 𝑗) → (𝑄‘𝑖) = (𝑄‘𝑗))
11	9	ffvelcdmda 7104	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
12	11	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘𝑖) ∈ ℝ)
13	9	ffvelcdmda 7104	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑀)) → (𝑄‘𝑘) ∈ ℝ)
14	13	ad4ant14 752	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0...𝑀)) → (𝑄‘𝑘) ∈ ℝ)
15	14	adantllr 719	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0...𝑀)) → (𝑄‘𝑘) ∈ ℝ)
16		eleq1w 2822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑘 → (𝑖 ∈ (0..^𝑀) ↔ 𝑘 ∈ (0..^𝑀)))
17	16	anbi2d 630	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑘 → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ 𝑘 ∈ (0..^𝑀))))
18		fveq2 6907	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑘 → (𝑄‘𝑖) = (𝑄‘𝑘))
19		oveq1 7438	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑘 → (𝑖 + 1) = (𝑘 + 1))
20	19	fveq2d 6911	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑘 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑘 + 1)))
21	18, 20	breq12d 5161	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑘 → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘𝑘) < (𝑄‘(𝑘 + 1))))
22	17, 21	imbi12d 344	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑘 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑 ∧ 𝑘 ∈ (0..^𝑀)) → (𝑄‘𝑘) < (𝑄‘(𝑘 + 1)))))
23	6	simprrd 774	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
24	23	r19.21bi 3249	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
25	22, 24	chvarvv 1996	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑀)) → (𝑄‘𝑘) < (𝑄‘(𝑘 + 1)))
26	25	ad4ant14 752	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑄‘𝑘) < (𝑄‘(𝑘 + 1)))
27	26	adantllr 719	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑄‘𝑘) < (𝑄‘(𝑘 + 1)))
28		simpllr 776	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 ∈ (0...𝑀))
29		simplr 769	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ (0...𝑀))
30		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
31	15, 27, 28, 29, 30	monoords 45248	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘𝑖) < (𝑄‘𝑗))
32	12, 31	ltned 11395	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘𝑖) ≠ (𝑄‘𝑗))
33	32	neneqd 2943	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → ¬ (𝑄‘𝑖) = (𝑄‘𝑗))
34	33	adantlr 715	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ 𝑖 < 𝑗) → ¬ (𝑄‘𝑖) = (𝑄‘𝑗))
35		simpll 767	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)))
36		elfzelz 13561	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
37	36	zred 12720	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
38	37	ad3antlr 731	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑗 ∈ ℝ)
39		elfzelz 13561	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
40	39	zred 12720	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℝ)
41	40	ad4antlr 733	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑖 ∈ ℝ)
42		neqne 2946	. . . . . . . . . . . 12 ⊢ (¬ 𝑖 = 𝑗 → 𝑖 ≠ 𝑗)
43	42	necomd 2994	. . . . . . . . . . 11 ⊢ (¬ 𝑖 = 𝑗 → 𝑗 ≠ 𝑖)
44	43	ad2antlr 727	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑗 ≠ 𝑖)
45		simpr 484	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → ¬ 𝑖 < 𝑗)
46	38, 41, 44, 45	lttri5d 45250	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑗 < 𝑖)
47	9	ffvelcdmda 7104	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑄‘𝑗) ∈ ℝ)
48	47	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄‘𝑗) ∈ ℝ)
49	48	adantllr 719	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄‘𝑗) ∈ ℝ)
50		simp-4l 783	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) ∧ 𝑘 ∈ (0...𝑀)) → 𝜑)
51	50, 13	sylancom 588	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) ∧ 𝑘 ∈ (0...𝑀)) → (𝑄‘𝑘) ∈ ℝ)
52		simp-4l 783	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) ∧ 𝑘 ∈ (0..^𝑀)) → 𝜑)
53	52, 25	sylancom 588	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑄‘𝑘) < (𝑄‘(𝑘 + 1)))
54		simplr 769	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → 𝑗 ∈ (0...𝑀))
55		simpllr 776	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → 𝑖 ∈ (0...𝑀))
56		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → 𝑗 < 𝑖)
57	51, 53, 54, 55, 56	monoords 45248	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄‘𝑗) < (𝑄‘𝑖))
58	49, 57	gtned 11394	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄‘𝑖) ≠ (𝑄‘𝑗))
59	58	neneqd 2943	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → ¬ (𝑄‘𝑖) = (𝑄‘𝑗))
60	35, 46, 59	syl2anc 584	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → ¬ (𝑄‘𝑖) = (𝑄‘𝑗))
61	34, 60	pm2.61dan 813	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) → ¬ (𝑄‘𝑖) = (𝑄‘𝑗))
62	61	adantlr 715	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = (𝑄‘𝑗)) ∧ ¬ 𝑖 = 𝑗) → ¬ (𝑄‘𝑖) = (𝑄‘𝑗))
63	10, 62	condan 818	. . . . 5 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = (𝑄‘𝑗)) → 𝑖 = 𝑗)
64	63	ex 412	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑄‘𝑖) = (𝑄‘𝑗) → 𝑖 = 𝑗))
65	64	ralrimiva 3144	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ∀𝑗 ∈ (0...𝑀)((𝑄‘𝑖) = (𝑄‘𝑗) → 𝑖 = 𝑗))
66	65	ralrimiva 3144	. 2 ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)((𝑄‘𝑖) = (𝑄‘𝑗) → 𝑖 = 𝑗))
67		dff13 7275	. 2 ⊢ (𝑄:(0...𝑀)–1-1→ℝ ↔ (𝑄:(0...𝑀)⟶ℝ ∧ ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)((𝑄‘𝑖) = (𝑄‘𝑗) → 𝑖 = 𝑗)))
68	9, 66, 67	sylanbrc 583	1 ⊢ (𝜑 → 𝑄:(0...𝑀)–1-1→ℝ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  {crab 3433   class class class wbr 5148   ↦ cmpt 5231  ⟶wf 6559  –1-1→wf1 6560  ‘cfv 6563  (class class class)co 7431   ↑_m cmap 8865  ℝcr 11152  0cc0 11153  1c1 11154   + caddc 11156   < clt 11293  ℕcn 12264  ...cfz 13544  ..^cfzo 13691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692

This theorem is referenced by:  fourierdlem50  46112