Theorem fourierdlem34 46150

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 46118	. . . . . 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 8868	. . 3 ⊢ (𝑄 ∈ (ℝ ↑m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
9	7, 8	syl 17	. 2 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
10		simplr 768	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄‘𝑖) = (𝑄‘𝑗)) ∧ ¬ 𝑖 = 𝑗) → (𝑄‘𝑖) = (𝑄‘𝑗))
11	9	ffvelcdmda 7079	. . . . . . . . . . . 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 7079	. . . . . . . . . . . . . 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 2818	. . . . . . . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑘 → (𝑄‘𝑖) = (𝑄‘𝑘))
19		oveq1 7417	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑘 → (𝑖 + 1) = (𝑘 + 1))
20	19	fveq2d 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑘 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑘 + 1)))
21	18, 20	breq12d 5137	. . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
24	23	r19.21bi 3238	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
25	22, 24	chvarvv 1989	. . . . . . . . . . . . . 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 775	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 ∈ (0...𝑀))
29		simplr 768	. . . . . . . . . . . 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 45306	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘𝑖) < (𝑄‘𝑗))
32	12, 31	ltned 11376	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘𝑖) ≠ (𝑄‘𝑗))
33	32	neneqd 2938	. . . . . . . . 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 766	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)))
36		elfzelz 13546	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
37	36	zred 12702	. . . . . . . . . . 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 13546	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
40	39	zred 12702	. . . . . . . . . . 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 2941	. . . . . . . . . . . 12 ⊢ (¬ 𝑖 = 𝑗 → 𝑖 ≠ 𝑗)
43	42	necomd 2988	. . . . . . . . . . 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 45308	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑗 < 𝑖)
47	9	ffvelcdmda 7079	. . . . . . . . . . . . 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 782	. . . . . . . . . . . . 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 782	. . . . . . . . . . . . 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 768	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → 𝑗 ∈ (0...𝑀))
55		simpllr 775	. . . . . . . . . . . 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 45306	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄‘𝑗) < (𝑄‘𝑖))
58	49, 57	gtned 11375	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄‘𝑖) ≠ (𝑄‘𝑗))
59	58	neneqd 2938	. . . . . . . . 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 812	. . . . . . 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 817	. . . . 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 3133	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ∀𝑗 ∈ (0...𝑀)((𝑄‘𝑖) = (𝑄‘𝑗) → 𝑖 = 𝑗))
66	65	ralrimiva 3133	. 2 ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)((𝑄‘𝑖) = (𝑄‘𝑗) → 𝑖 = 𝑗))
67		dff13 7252	. 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 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  {crab 3420   class class class wbr 5124   ↦ cmpt 5206  ⟶wf 6532  –1-1→wf1 6533  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8845  ℝcr 11133  0cc0 11134  1c1 11135   + caddc 11137   < clt 11274  ℕcn 12245  ...cfz 13529  ..^cfzo 13676

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 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-n0 12507  df-z 12594  df-uz 12858  df-fz 13530  df-fzo 13677

This theorem is referenced by:  fourierdlem50  46165