Theorem fvmptnn04if 22963

Description: The function values of a mapping from the nonnegative integers with four distinct cases. (Contributed by AV, 10-Nov-2019.)

Hypotheses

Ref	Expression
fvmptnn04if.g	⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))
fvmptnn04if.s	⊢ (𝜑 → 𝑆 ∈ ℕ)
fvmptnn04if.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)
fvmptnn04if.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
fvmptnn04if.a	⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐴)
fvmptnn04if.b	⊢ ((𝜑 ∧ 0 < 𝑁 ∧ 𝑁 < 𝑆) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐵)
fvmptnn04if.c	⊢ ((𝜑 ∧ 𝑁 = 𝑆) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐶)
fvmptnn04if.d	⊢ ((𝜑 ∧ 𝑆 < 𝑁) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐷)

Assertion

Ref	Expression
fvmptnn04if	⊢ (𝜑 → (𝐺‘𝑁) = 𝑌)

Distinct variable groups:   𝑛,𝑁   𝑆,𝑛

Allowed substitution hints:   𝜑(𝑛)   𝐴(𝑛)   𝐵(𝑛)   𝐶(𝑛)   𝐷(𝑛)   𝐺(𝑛)   𝑉(𝑛)   𝑌(𝑛)

Proof of Theorem fvmptnn04if

Step	Hyp	Ref	Expression
1		fvmptnn04if.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
2		csbif 4541	. . . . 5 ⊢ ⦋𝑁 / 𝑛⦌if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))) = if([𝑁 / 𝑛]𝑛 = 0, ⦋𝑁 / 𝑛⦌𝐴, ⦋𝑁 / 𝑛⦌if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵)))
3		eqsbc1 3793	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑛 = 0 ↔ 𝑁 = 0))
4	1, 3	syl 18	. . . . . 6 ⊢ (𝜑 → ([𝑁 / 𝑛]𝑛 = 0 ↔ 𝑁 = 0))
5		csbif 4541	. . . . . . 7 ⊢ ⦋𝑁 / 𝑛⦌if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵)) = if([𝑁 / 𝑛]𝑛 = 𝑆, ⦋𝑁 / 𝑛⦌𝐶, ⦋𝑁 / 𝑛⦌if(𝑆 < 𝑛, 𝐷, 𝐵))
6		eqsbc1 3793	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑛 = 𝑆 ↔ 𝑁 = 𝑆))
7	1, 6	syl 18	. . . . . . . 8 ⊢ (𝜑 → ([𝑁 / 𝑛]𝑛 = 𝑆 ↔ 𝑁 = 𝑆))
8		csbif 4541	. . . . . . . . 9 ⊢ ⦋𝑁 / 𝑛⦌if(𝑆 < 𝑛, 𝐷, 𝐵) = if([𝑁 / 𝑛]𝑆 < 𝑛, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵)
9		sbcbr2g 5162	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑆 < 𝑛 ↔ 𝑆 < ⦋𝑁 / 𝑛⦌𝑛))
10	1, 9	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ([𝑁 / 𝑛]𝑆 < 𝑛 ↔ 𝑆 < ⦋𝑁 / 𝑛⦌𝑛))
11		csbvarg 4391	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → ⦋𝑁 / 𝑛⦌𝑛 = 𝑁)
12	1, 11	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ⦋𝑁 / 𝑛⦌𝑛 = 𝑁)
13	12	breq2d 5116	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 < ⦋𝑁 / 𝑛⦌𝑛 ↔ 𝑆 < 𝑁))
14	10, 13	bitrd 282	. . . . . . . . . 10 ⊢ (𝜑 → ([𝑁 / 𝑛]𝑆 < 𝑛 ↔ 𝑆 < 𝑁))
15	14	ifbid 4507	. . . . . . . . 9 ⊢ (𝜑 → if([𝑁 / 𝑛]𝑆 < 𝑛, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵) = if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵))
16	8, 15	eqtrid 2812	. . . . . . . 8 ⊢ (𝜑 → ⦋𝑁 / 𝑛⦌if(𝑆 < 𝑛, 𝐷, 𝐵) = if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵))
17	7, 16	ifbieq2d 4510	. . . . . . 7 ⊢ (𝜑 → if([𝑁 / 𝑛]𝑛 = 𝑆, ⦋𝑁 / 𝑛⦌𝐶, ⦋𝑁 / 𝑛⦌if(𝑆 < 𝑛, 𝐷, 𝐵)) = if(𝑁 = 𝑆, ⦋𝑁 / 𝑛⦌𝐶, if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵)))
18	5, 17	eqtrid 2812	. . . . . 6 ⊢ (𝜑 → ⦋𝑁 / 𝑛⦌if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵)) = if(𝑁 = 𝑆, ⦋𝑁 / 𝑛⦌𝐶, if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵)))
19	4, 18	ifbieq2d 4510	. . . . 5 ⊢ (𝜑 → if([𝑁 / 𝑛]𝑛 = 0, ⦋𝑁 / 𝑛⦌𝐴, ⦋𝑁 / 𝑛⦌if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))) = if(𝑁 = 0, ⦋𝑁 / 𝑛⦌𝐴, if(𝑁 = 𝑆, ⦋𝑁 / 𝑛⦌𝐶, if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵))))
20	2, 19	eqtrid 2812	. . . 4 ⊢ (𝜑 → ⦋𝑁 / 𝑛⦌if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))) = if(𝑁 = 0, ⦋𝑁 / 𝑛⦌𝐴, if(𝑁 = 𝑆, ⦋𝑁 / 𝑛⦌𝐶, if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵))))
21		fvmptnn04if.a	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐴)
22		fvmptnn04if.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
23	22	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑌 ∈ 𝑉)
24	21, 23	eqeltrrd 2866	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 = 0) → ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉)
25		fvmptnn04if.c	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 = 𝑆) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐶)
26	25	eqcomd 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 = 𝑆) → ⦋𝑁 / 𝑛⦌𝐶 = 𝑌)
27	26	adantlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → ⦋𝑁 / 𝑛⦌𝐶 = 𝑌)
28	22	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → 𝑌 ∈ 𝑉)
29	27, 28	eqeltrd 2865	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉)
30		fvmptnn04if.d	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑆 < 𝑁) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐷)
31	30	eqcomd 2771	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑆 < 𝑁) → ⦋𝑁 / 𝑛⦌𝐷 = 𝑌)
32	31	ad4ant14 764	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → ⦋𝑁 / 𝑛⦌𝐷 = 𝑌)
33	22	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → 𝑌 ∈ 𝑉)
34	32, 33	eqeltrd 2865	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → ⦋𝑁 / 𝑛⦌𝐷 ∈ 𝑉)
35		simplll 786	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝜑)
36		anass 473	. . . . . . . . . . . . 13 ⊢ (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) ↔ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)))
37	36	bicomi 227	. . . . . . . . . . . 12 ⊢ ((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) ↔ ((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁))
38	37	bianassc 655	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) ↔ (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ∧ ¬ 𝑆 < 𝑁))
39		an32 658	. . . . . . . . . . . . 13 ⊢ (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ↔ ((¬ 𝑁 = 0 ∧ 𝜑) ∧ ¬ 𝑁 = 𝑆))
40		ancom 465	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑁 = 0 ∧ 𝜑) ↔ (𝜑 ∧ ¬ 𝑁 = 0))
41	40	anbi1i 635	. . . . . . . . . . . . 13 ⊢ (((¬ 𝑁 = 0 ∧ 𝜑) ∧ ¬ 𝑁 = 𝑆) ↔ ((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆))
42	39, 41	bitri 278	. . . . . . . . . . . 12 ⊢ (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ↔ ((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆))
43	42	anbi1i 635	. . . . . . . . . . 11 ⊢ ((((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ∧ ¬ 𝑆 < 𝑁) ↔ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁))
44	38, 43	bitri 278	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) ↔ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁))
45		df-ne 2961	. . . . . . . . . . . . 13 ⊢ (𝑁 ≠ 0 ↔ ¬ 𝑁 = 0)
46		elnnne0 12506	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0))
47		nngt0 12255	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
48	46, 47	sylbir 238	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0) → 0 < 𝑁)
49	48	expcom 418	. . . . . . . . . . . . 13 ⊢ (𝑁 ≠ 0 → (𝑁 ∈ ℕ0 → 0 < 𝑁))
50	45, 49	sylbir 238	. . . . . . . . . . . 12 ⊢ (¬ 𝑁 = 0 → (𝑁 ∈ ℕ0 → 0 < 𝑁))
51	50	adantr 485	. . . . . . . . . . 11 ⊢ ((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) → (𝑁 ∈ ℕ0 → 0 < 𝑁))
52	1, 51	mpan9 515	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 0 < 𝑁)
53	44, 52	sylbir 238	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 0 < 𝑁)
54	1	nn0red 12554	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℝ)
55	54	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑁 ∈ ℝ)
56		fvmptnn04if.s	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ ℕ)
57	56	nnred 12236	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ ℝ)
58	57	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑆 ∈ ℝ)
59	54, 57	lenltd 11344	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 ≤ 𝑆 ↔ ¬ 𝑆 < 𝑁))
60	59	biimprd 251	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (¬ 𝑆 < 𝑁 → 𝑁 ≤ 𝑆))
61	60	adantld 495	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁) → 𝑁 ≤ 𝑆))
62	61	adantld 495	. . . . . . . . . . . 12 ⊢ (𝜑 → ((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) → 𝑁 ≤ 𝑆))
63	62	imp 411	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑁 ≤ 𝑆)
64		nesym 3016	. . . . . . . . . . . . 13 ⊢ (𝑆 ≠ 𝑁 ↔ ¬ 𝑁 = 𝑆)
65	64	biranri 510	. . . . . . . . . . . 12 ⊢ ((¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁) → 𝑆 ≠ 𝑁)
66	65	ad2antll 741	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑆 ≠ 𝑁)
67	55, 58, 63, 66	leneltd 11352	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑁 < 𝑆)
68	44, 67	sylbir 238	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝑁 < 𝑆)
69		fvmptnn04if.b	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < 𝑁 ∧ 𝑁 < 𝑆) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐵)
70	69	eqcomd 2771	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < 𝑁 ∧ 𝑁 < 𝑆) → ⦋𝑁 / 𝑛⦌𝐵 = 𝑌)
71	35, 53, 68, 70	syl3anc 1394	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → ⦋𝑁 / 𝑛⦌𝐵 = 𝑌)
72	22	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝑌 ∈ 𝑉)
73	71, 72	eqeltrd 2865	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉)
74	34, 73	ifclda 4519	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) → if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵) ∈ 𝑉)
75	29, 74	ifclda 4519	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 = 0) → if(𝑁 = 𝑆, ⦋𝑁 / 𝑛⦌𝐶, if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵)) ∈ 𝑉)
76	24, 75	ifclda 4519	. . . 4 ⊢ (𝜑 → if(𝑁 = 0, ⦋𝑁 / 𝑛⦌𝐴, if(𝑁 = 𝑆, ⦋𝑁 / 𝑛⦌𝐶, if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵))) ∈ 𝑉)
77	20, 76	eqeltrd 2865	. . 3 ⊢ (𝜑 → ⦋𝑁 / 𝑛⦌if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))) ∈ 𝑉)
78		fvmptnn04if.g	. . . 4 ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))
79	78	fvmpts 6983	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ ⦋𝑁 / 𝑛⦌if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))) ∈ 𝑉) → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))
80	1, 77, 79	syl2anc 595	. 2 ⊢ (𝜑 → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))
81	21	eqcomd 2771	. . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → ⦋𝑁 / 𝑛⦌𝐴 = 𝑌)
82	32, 71	ifeqda 4520	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) → if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵) = 𝑌)
83	27, 82	ifeqda 4520	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑁 = 0) → if(𝑁 = 𝑆, ⦋𝑁 / 𝑛⦌𝐶, if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵)) = 𝑌)
84	81, 83	ifeqda 4520	. 2 ⊢ (𝜑 → if(𝑁 = 0, ⦋𝑁 / 𝑛⦌𝐴, if(𝑁 = 𝑆, ⦋𝑁 / 𝑛⦌𝐶, if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵))) = 𝑌)
85	80, 20, 84	3eqtrd 2804	1 ⊢ (𝜑 → (𝐺‘𝑁) = 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  [wsbc 3747  ⦋csb 3855  ifcif 4483   class class class wbr 5104   ↦ cmpt 5185  ‘cfv 6525  ℝcr 11087  0cc0 11088   < clt 11231   ≤ cle 11232  ℕcn 12221  ℕ₀cn0 12492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12222  df-n0 12493

This theorem is referenced by:  fvmptnn04ifa  22964  fvmptnn04ifb  22965  fvmptnn04ifc  22966  fvmptnn04ifd  22967