Theorem fvmptnn04if 22987

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 4546	. . . . 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 3791	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑛 = 0 ↔ 𝑁 = 0))
4	1, 3	syl 18	. . . . . 6 ⊢ (𝜑 → ([𝑁 / 𝑛]𝑛 = 0 ↔ 𝑁 = 0))
5		csbif 4546	. . . . . . 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 3791	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑛 = 𝑆 ↔ 𝑁 = 𝑆))
7	1, 6	syl 18	. . . . . . . 8 ⊢ (𝜑 → ([𝑁 / 𝑛]𝑛 = 𝑆 ↔ 𝑁 = 𝑆))
8		csbif 4546	. . . . . . . . 9 ⊢ ⦋𝑁 / 𝑛⦌if(𝑆 < 𝑛, 𝐷, 𝐵) = if([𝑁 / 𝑛]𝑆 < 𝑛, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵)
9		sbcbr2g 5170	. . . . . . . . . . . 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 4400	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → ⦋𝑁 / 𝑛⦌𝑛 = 𝑁)
12	1, 11	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ⦋𝑁 / 𝑛⦌𝑛 = 𝑁)
13	12	breq2d 5122	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 < ⦋𝑁 / 𝑛⦌𝑛 ↔ 𝑆 < 𝑁))
14	10, 13	bitrd 282	. . . . . . . . . 10 ⊢ (𝜑 → ([𝑁 / 𝑛]𝑆 < 𝑛 ↔ 𝑆 < 𝑁))
15	14	ifbid 4512	. . . . . . . . 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 2810	. . . . . . . 8 ⊢ (𝜑 → ⦋𝑁 / 𝑛⦌if(𝑆 < 𝑛, 𝐷, 𝐵) = if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵))
17	7, 16	ifbieq2d 4515	. . . . . . 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 2810	. . . . . 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 4515	. . . . 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 2810	. . . 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 2864	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 = 0) → ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉)
25		fvmptnn04if.c	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 = 𝑆) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐶)
26	25	eqcomd 2769	. . . . . . . 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 2863	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉)
30		fvmptnn04if.d	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑆 < 𝑁) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐷)
31	30	eqcomd 2769	. . . . . . . . 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 2863	. . . . . . 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 2959	. . . . . . . . . . . . 13 ⊢ (𝑁 ≠ 0 ↔ ¬ 𝑁 = 0)
46		elnnne0 12519	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0))
47		nngt0 12268	. . . . . . . . . . . . . . 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 12567	. . . . . . . . . . . 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 12249	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ ℝ)
58	57	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑆 ∈ ℝ)
59	54, 57	lenltd 11357	. . . . . . . . . . . . . . 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 3014	. . . . . . . . . . . . 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 11365	. . . . . . . . . 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 2769	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < 𝑁 ∧ 𝑁 < 𝑆) → ⦋𝑁 / 𝑛⦌𝐵 = 𝑌)
71	35, 53, 68, 70	syl3anc 1398	. . . . . . . 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 2863	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉)
74	34, 73	ifclda 4524	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) → if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵) ∈ 𝑉)
75	29, 74	ifclda 4524	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 = 0) → if(𝑁 = 𝑆, ⦋𝑁 / 𝑛⦌𝐶, if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵)) ∈ 𝑉)
76	24, 75	ifclda 4524	. . . 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 2863	. . 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 6995	. . 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 2769	. . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → ⦋𝑁 / 𝑛⦌𝐴 = 𝑌)
82	32, 71	ifeqda 4525	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) → if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵) = 𝑌)
83	27, 82	ifeqda 4525	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑁 = 0) → if(𝑁 = 𝑆, ⦋𝑁 / 𝑛⦌𝐶, if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵)) = 𝑌)
84	81, 83	ifeqda 4525	. 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 2802	1 ⊢ (𝜑 → (𝐺‘𝑁) = 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  [wsbc 3745  ⦋csb 3854  ifcif 4488   class class class wbr 5110   ↦ cmpt 5193  ‘cfv 6538  ℝcr 11100  0cc0 11101   < clt 11244   ≤ cle 11245  ℕcn 12234  ℕ₀cn0 12505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-n0 12506

This theorem is referenced by:  fvmptnn04ifa  22988  fvmptnn04ifb  22989  fvmptnn04ifc  22990  fvmptnn04ifd  22991