Theorem fvmptnn04if 22198

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 4543	. . . . 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 3788	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑛 = 0 ↔ 𝑁 = 0))
4	1, 3	syl 17	. . . . . 6 ⊢ (𝜑 → ([𝑁 / 𝑛]𝑛 = 0 ↔ 𝑁 = 0))
5		csbif 4543	. . . . . . 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 3788	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑛 = 𝑆 ↔ 𝑁 = 𝑆))
7	1, 6	syl 17	. . . . . . . 8 ⊢ (𝜑 → ([𝑁 / 𝑛]𝑛 = 𝑆 ↔ 𝑁 = 𝑆))
8		csbif 4543	. . . . . . . . 9 ⊢ ⦋𝑁 / 𝑛⦌if(𝑆 < 𝑛, 𝐷, 𝐵) = if([𝑁 / 𝑛]𝑆 < 𝑛, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵)
9		sbcbr2g 5163	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑆 < 𝑛 ↔ 𝑆 < ⦋𝑁 / 𝑛⦌𝑛))
10	1, 9	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ([𝑁 / 𝑛]𝑆 < 𝑛 ↔ 𝑆 < ⦋𝑁 / 𝑛⦌𝑛))
11		csbvarg 4391	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → ⦋𝑁 / 𝑛⦌𝑛 = 𝑁)
12	1, 11	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ⦋𝑁 / 𝑛⦌𝑛 = 𝑁)
13	12	breq2d 5117	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 < ⦋𝑁 / 𝑛⦌𝑛 ↔ 𝑆 < 𝑁))
14	10, 13	bitrd 278	. . . . . . . . . 10 ⊢ (𝜑 → ([𝑁 / 𝑛]𝑆 < 𝑛 ↔ 𝑆 < 𝑁))
15	14	ifbid 4509	. . . . . . . . 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 2788	. . . . . . . 8 ⊢ (𝜑 → ⦋𝑁 / 𝑛⦌if(𝑆 < 𝑛, 𝐷, 𝐵) = if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵))
17	7, 16	ifbieq2d 4512	. . . . . . 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 2788	. . . . . 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 4512	. . . . 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 2788	. . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑌 ∈ 𝑉)
24	21, 23	eqeltrrd 2839	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 = 0) → ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉)
25		fvmptnn04if.c	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 = 𝑆) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐶)
26	25	eqcomd 2742	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 = 𝑆) → ⦋𝑁 / 𝑛⦌𝐶 = 𝑌)
27	26	adantlr 713	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → ⦋𝑁 / 𝑛⦌𝐶 = 𝑌)
28	22	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → 𝑌 ∈ 𝑉)
29	27, 28	eqeltrd 2838	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉)
30		fvmptnn04if.d	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑆 < 𝑁) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐷)
31	30	eqcomd 2742	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑆 < 𝑁) → ⦋𝑁 / 𝑛⦌𝐷 = 𝑌)
32	31	ad4ant14 750	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → ⦋𝑁 / 𝑛⦌𝐷 = 𝑌)
33	22	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → 𝑌 ∈ 𝑉)
34	32, 33	eqeltrd 2838	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → ⦋𝑁 / 𝑛⦌𝐷 ∈ 𝑉)
35		simplll 773	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝜑)
36		anass 469	. . . . . . . . . . . . 13 ⊢ (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) ↔ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)))
37	36	bicomi 223	. . . . . . . . . . . 12 ⊢ ((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) ↔ ((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁))
38	37	bianassc 641	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) ↔ (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ∧ ¬ 𝑆 < 𝑁))
39		an32 644	. . . . . . . . . . . . 13 ⊢ (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ↔ ((¬ 𝑁 = 0 ∧ 𝜑) ∧ ¬ 𝑁 = 𝑆))
40		ancom 461	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑁 = 0 ∧ 𝜑) ↔ (𝜑 ∧ ¬ 𝑁 = 0))
41	40	anbi1i 624	. . . . . . . . . . . . 13 ⊢ (((¬ 𝑁 = 0 ∧ 𝜑) ∧ ¬ 𝑁 = 𝑆) ↔ ((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆))
42	39, 41	bitri 274	. . . . . . . . . . . 12 ⊢ (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ↔ ((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆))
43	42	anbi1i 624	. . . . . . . . . . 11 ⊢ ((((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ∧ ¬ 𝑆 < 𝑁) ↔ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁))
44	38, 43	bitri 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) ↔ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁))
45		df-ne 2944	. . . . . . . . . . . . 13 ⊢ (𝑁 ≠ 0 ↔ ¬ 𝑁 = 0)
46		elnnne0 12427	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0))
47		nngt0 12184	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
48	46, 47	sylbir 234	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0) → 0 < 𝑁)
49	48	expcom 414	. . . . . . . . . . . . 13 ⊢ (𝑁 ≠ 0 → (𝑁 ∈ ℕ0 → 0 < 𝑁))
50	45, 49	sylbir 234	. . . . . . . . . . . 12 ⊢ (¬ 𝑁 = 0 → (𝑁 ∈ ℕ0 → 0 < 𝑁))
51	50	adantr 481	. . . . . . . . . . 11 ⊢ ((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) → (𝑁 ∈ ℕ0 → 0 < 𝑁))
52	1, 51	mpan9 507	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 0 < 𝑁)
53	44, 52	sylbir 234	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 0 < 𝑁)
54	1	nn0red 12474	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℝ)
55	54	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑁 ∈ ℝ)
56		fvmptnn04if.s	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ ℕ)
57	56	nnred 12168	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ ℝ)
58	57	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑆 ∈ ℝ)
59	54, 57	lenltd 11301	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 ≤ 𝑆 ↔ ¬ 𝑆 < 𝑁))
60	59	biimprd 247	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (¬ 𝑆 < 𝑁 → 𝑁 ≤ 𝑆))
61	60	adantld 491	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁) → 𝑁 ≤ 𝑆))
62	61	adantld 491	. . . . . . . . . . . 12 ⊢ (𝜑 → ((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) → 𝑁 ≤ 𝑆))
63	62	imp 407	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑁 ≤ 𝑆)
64		nesym 3000	. . . . . . . . . . . . . 14 ⊢ (𝑆 ≠ 𝑁 ↔ ¬ 𝑁 = 𝑆)
65	64	biimpri 227	. . . . . . . . . . . . 13 ⊢ (¬ 𝑁 = 𝑆 → 𝑆 ≠ 𝑁)
66	65	adantr 481	. . . . . . . . . . . 12 ⊢ ((¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁) → 𝑆 ≠ 𝑁)
67	66	ad2antll 727	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑆 ≠ 𝑁)
68	55, 58, 63, 67	leneltd 11309	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑁 < 𝑆)
69	44, 68	sylbir 234	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝑁 < 𝑆)
70		fvmptnn04if.b	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 0 < 𝑁 ∧ 𝑁 < 𝑆) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐵)
71	70	eqcomd 2742	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < 𝑁 ∧ 𝑁 < 𝑆) → ⦋𝑁 / 𝑛⦌𝐵 = 𝑌)
72	35, 53, 69, 71	syl3anc 1371	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → ⦋𝑁 / 𝑛⦌𝐵 = 𝑌)
73	22	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝑌 ∈ 𝑉)
74	72, 73	eqeltrd 2838	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉)
75	34, 74	ifclda 4521	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) → if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵) ∈ 𝑉)
76	29, 75	ifclda 4521	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 = 0) → if(𝑁 = 𝑆, ⦋𝑁 / 𝑛⦌𝐶, if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵)) ∈ 𝑉)
77	24, 76	ifclda 4521	. . . 4 ⊢ (𝜑 → if(𝑁 = 0, ⦋𝑁 / 𝑛⦌𝐴, if(𝑁 = 𝑆, ⦋𝑁 / 𝑛⦌𝐶, if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵))) ∈ 𝑉)
78	20, 77	eqeltrd 2838	. . 3 ⊢ (𝜑 → ⦋𝑁 / 𝑛⦌if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))) ∈ 𝑉)
79		fvmptnn04if.g	. . . 4 ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))
80	79	fvmpts 6951	. . 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))))
81	1, 78, 80	syl2anc 584	. 2 ⊢ (𝜑 → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))
82	21	eqcomd 2742	. . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → ⦋𝑁 / 𝑛⦌𝐴 = 𝑌)
83	32, 72	ifeqda 4522	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) → if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵) = 𝑌)
84	27, 83	ifeqda 4522	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑁 = 0) → if(𝑁 = 𝑆, ⦋𝑁 / 𝑛⦌𝐶, if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵)) = 𝑌)
85	82, 84	ifeqda 4522	. 2 ⊢ (𝜑 → if(𝑁 = 0, ⦋𝑁 / 𝑛⦌𝐴, if(𝑁 = 𝑆, ⦋𝑁 / 𝑛⦌𝐶, if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵))) = 𝑌)
86	81, 20, 85	3eqtrd 2780	1 ⊢ (𝜑 → (𝐺‘𝑁) = 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  [wsbc 3739  ⦋csb 3855  ifcif 4486   class class class wbr 5105   ↦ cmpt 5188  ‘cfv 6496  ℝcr 11050  0cc0 11051   < clt 11189   ≤ cle 11190  ℕcn 12153  ℕ₀cn0 12413

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414

This theorem is referenced by:  fvmptnn04ifa  22199  fvmptnn04ifb  22200  fvmptnn04ifc  22201  fvmptnn04ifd  22202