Theorem fourierdlem34 46161

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 46129	. . . . . 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 8890	. . 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 7103	. . . . . . . . . . . 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 7103	. . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑘 → (𝑄‘𝑖) = (𝑄‘𝑘))
19		oveq1 7439	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑘 → (𝑖 + 1) = (𝑘 + 1))
20	19	fveq2d 6909	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑘 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑘 + 1)))
21	18, 20	breq12d 5155	. . . . . . . . . . . . . . . 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 3250	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
25	22, 24	chvarvv 1997	. . . . . . . . . . . . . 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 45314	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘𝑖) < (𝑄‘𝑗))
32	12, 31	ltned 11398	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘𝑖) ≠ (𝑄‘𝑗))
33	32	neneqd 2944	. . . . . . . . 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 13565	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
37	36	zred 12724	. . . . . . . . . . 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 13565	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
40	39	zred 12724	. . . . . . . . . . 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 2947	. . . . . . . . . . . 12 ⊢ (¬ 𝑖 = 𝑗 → 𝑖 ≠ 𝑗)
43	42	necomd 2995	. . . . . . . . . . 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 45316	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑗 < 𝑖)
47	9	ffvelcdmda 7103	. . . . . . . . . . . . 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 45314	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄‘𝑗) < (𝑄‘𝑖))
58	49, 57	gtned 11397	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄‘𝑖) ≠ (𝑄‘𝑗))
59	58	neneqd 2944	. . . . . . . . 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 3145	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ∀𝑗 ∈ (0...𝑀)((𝑄‘𝑖) = (𝑄‘𝑗) → 𝑖 = 𝑗))
66	65	ralrimiva 3145	. 2 ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)((𝑄‘𝑖) = (𝑄‘𝑗) → 𝑖 = 𝑗))
67		dff13 7276	. 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 1539   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  {crab 3435   class class class wbr 5142   ↦ cmpt 5224  ⟶wf 6556  –1-1→wf1 6557  ‘cfv 6560  (class class class)co 7432   ↑_m cmap 8867  ℝcr 11155  0cc0 11156  1c1 11157   + caddc 11159   < clt 11296  ℕcn 12267  ...cfz 13548  ..^cfzo 13695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-n0 12529  df-z 12616  df-uz 12880  df-fz 13549  df-fzo 13696

This theorem is referenced by:  fourierdlem50  46176