Theorem fourierdlem34 46590

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 46558	. . . . . 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 8790	. . 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 7031	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
12	11	ad2antrr 727	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘𝑖) ∈ ℝ)
13	9	ffvelcdmda 7031	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑀)) → (𝑄‘𝑘) ∈ ℝ)
14	13	ad4ant14 753	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0...𝑀)) → (𝑄‘𝑘) ∈ ℝ)
15	14	adantllr 720	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0...𝑀)) → (𝑄‘𝑘) ∈ ℝ)
16		eleq1w 2820	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑘 → (𝑖 ∈ (0..^𝑀) ↔ 𝑘 ∈ (0..^𝑀)))
17	16	anbi2d 631	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑘 → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ 𝑘 ∈ (0..^𝑀))))
18		fveq2 6835	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑘 → (𝑄‘𝑖) = (𝑄‘𝑘))
19		oveq1 7368	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑘 → (𝑖 + 1) = (𝑘 + 1))
20	19	fveq2d 6839	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑘 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑘 + 1)))
21	18, 20	breq12d 5099	. . . . . . . . . . . . . . . 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 3230	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
25	22, 24	chvarvv 1991	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑀)) → (𝑄‘𝑘) < (𝑄‘(𝑘 + 1)))
26	25	ad4ant14 753	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑄‘𝑘) < (𝑄‘(𝑘 + 1)))
27	26	adantllr 720	. . . . . . . . . . . 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 45751	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄‘𝑖) < (𝑄‘𝑗))
32	12, 31	ltned 11276	. . . . . . . . . 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 716	. . . . . . . 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 13472	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
37	36	zred 12627	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
38	37	ad3antlr 732	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑗 ∈ ℝ)
39		elfzelz 13472	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
40	39	zred 12627	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℝ)
41	40	ad4antlr 734	. . . . . . . . . 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 728	. . . . . . . . . 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 45753	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑗 < 𝑖)
47	9	ffvelcdmda 7031	. . . . . . . . . . . . 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 720	. . . . . . . . . . 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 589	. . . . . . . . . . . 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 589	. . . . . . . . . . . 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 45751	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄‘𝑗) < (𝑄‘𝑖))
58	49, 57	gtned 11275	. . . . . . . . . 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 585	. . . . . . . 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 716	. . . . . 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 3130	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ∀𝑗 ∈ (0...𝑀)((𝑄‘𝑖) = (𝑄‘𝑗) → 𝑖 = 𝑗))
66	65	ralrimiva 3130	. 2 ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)((𝑄‘𝑖) = (𝑄‘𝑗) → 𝑖 = 𝑗))
67		dff13 7203	. 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 584	1 ⊢ (𝜑 → 𝑄:(0...𝑀)–1-1→ℝ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  {crab 3390   class class class wbr 5086   ↦ cmpt 5167  ⟶wf 6489  –1-1→wf1 6490  ‘cfv 6493  (class class class)co 7361   ↑_m cmap 8767  ℝcr 11031  0cc0 11032  1c1 11033   + caddc 11035   < clt 11173  ℕcn 12168  ...cfz 13455  ..^cfzo 13602

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 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-n0 12432  df-z 12519  df-uz 12783  df-fz 13456  df-fzo 13603

This theorem is referenced by:  fourierdlem50  46605