Theorem fvmptnn04if 22824

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 4525	. . . . 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 3776	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑛 = 0 ↔ 𝑁 = 0))
4	1, 3	syl 17	. . . . . 6 ⊢ (𝜑 → ([𝑁 / 𝑛]𝑛 = 0 ↔ 𝑁 = 0))
5		csbif 4525	. . . . . . 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 3776	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → ([𝑁 / 𝑛]𝑛 = 𝑆 ↔ 𝑁 = 𝑆))
7	1, 6	syl 17	. . . . . . . 8 ⊢ (𝜑 → ([𝑁 / 𝑛]𝑛 = 𝑆 ↔ 𝑁 = 𝑆))
8		csbif 4525	. . . . . . . . 9 ⊢ ⦋𝑁 / 𝑛⦌if(𝑆 < 𝑛, 𝐷, 𝐵) = if([𝑁 / 𝑛]𝑆 < 𝑛, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵)
9		sbcbr2g 5144	. . . . . . . . . . . 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 4375	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → ⦋𝑁 / 𝑛⦌𝑛 = 𝑁)
12	1, 11	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ⦋𝑁 / 𝑛⦌𝑛 = 𝑁)
13	12	breq2d 5098	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 < ⦋𝑁 / 𝑛⦌𝑛 ↔ 𝑆 < 𝑁))
14	10, 13	bitrd 279	. . . . . . . . . 10 ⊢ (𝜑 → ([𝑁 / 𝑛]𝑆 < 𝑛 ↔ 𝑆 < 𝑁))
15	14	ifbid 4491	. . . . . . . . 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 2784	. . . . . . . 8 ⊢ (𝜑 → ⦋𝑁 / 𝑛⦌if(𝑆 < 𝑛, 𝐷, 𝐵) = if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵))
17	7, 16	ifbieq2d 4494	. . . . . . 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 2784	. . . . . 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 4494	. . . . 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 2784	. . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑌 ∈ 𝑉)
24	21, 23	eqeltrrd 2838	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 = 0) → ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉)
25		fvmptnn04if.c	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 = 𝑆) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐶)
26	25	eqcomd 2743	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 = 𝑆) → ⦋𝑁 / 𝑛⦌𝐶 = 𝑌)
27	26	adantlr 716	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → ⦋𝑁 / 𝑛⦌𝐶 = 𝑌)
28	22	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → 𝑌 ∈ 𝑉)
29	27, 28	eqeltrd 2837	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ 𝑁 = 𝑆) → ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉)
30		fvmptnn04if.d	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑆 < 𝑁) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐷)
31	30	eqcomd 2743	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑆 < 𝑁) → ⦋𝑁 / 𝑛⦌𝐷 = 𝑌)
32	31	ad4ant14 753	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → ⦋𝑁 / 𝑛⦌𝐷 = 𝑌)
33	22	ad3antrrr 731	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → 𝑌 ∈ 𝑉)
34	32, 33	eqeltrd 2837	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ 𝑆 < 𝑁) → ⦋𝑁 / 𝑛⦌𝐷 ∈ 𝑉)
35		simplll 775	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝜑)
36		anass 468	. . . . . . . . . . . . 13 ⊢ (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) ↔ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)))
37	36	bicomi 224	. . . . . . . . . . . 12 ⊢ ((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) ↔ ((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁))
38	37	bianassc 644	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) ↔ (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ∧ ¬ 𝑆 < 𝑁))
39		an32 647	. . . . . . . . . . . . 13 ⊢ (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ↔ ((¬ 𝑁 = 0 ∧ 𝜑) ∧ ¬ 𝑁 = 𝑆))
40		ancom 460	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑁 = 0 ∧ 𝜑) ↔ (𝜑 ∧ ¬ 𝑁 = 0))
41	40	anbi1i 625	. . . . . . . . . . . . 13 ⊢ (((¬ 𝑁 = 0 ∧ 𝜑) ∧ ¬ 𝑁 = 𝑆) ↔ ((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆))
42	39, 41	bitri 275	. . . . . . . . . . . 12 ⊢ (((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ↔ ((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆))
43	42	anbi1i 625	. . . . . . . . . . 11 ⊢ ((((¬ 𝑁 = 0 ∧ ¬ 𝑁 = 𝑆) ∧ 𝜑) ∧ ¬ 𝑆 < 𝑁) ↔ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁))
44	38, 43	bitri 275	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) ↔ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁))
45		df-ne 2934	. . . . . . . . . . . . 13 ⊢ (𝑁 ≠ 0 ↔ ¬ 𝑁 = 0)
46		elnnne0 12442	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0))
47		nngt0 12199	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
48	46, 47	sylbir 235	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0) → 0 < 𝑁)
49	48	expcom 413	. . . . . . . . . . . . 13 ⊢ (𝑁 ≠ 0 → (𝑁 ∈ ℕ0 → 0 < 𝑁))
50	45, 49	sylbir 235	. . . . . . . . . . . 12 ⊢ (¬ 𝑁 = 0 → (𝑁 ∈ ℕ0 → 0 < 𝑁))
51	50	adantr 480	. . . . . . . . . . 11 ⊢ ((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) → (𝑁 ∈ ℕ0 → 0 < 𝑁))
52	1, 51	mpan9 506	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 0 < 𝑁)
53	44, 52	sylbir 235	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 0 < 𝑁)
54	1	nn0red 12490	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℝ)
55	54	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑁 ∈ ℝ)
56		fvmptnn04if.s	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ ℕ)
57	56	nnred 12180	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ ℝ)
58	57	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑆 ∈ ℝ)
59	54, 57	lenltd 11283	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 ≤ 𝑆 ↔ ¬ 𝑆 < 𝑁))
60	59	biimprd 248	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (¬ 𝑆 < 𝑁 → 𝑁 ≤ 𝑆))
61	60	adantld 490	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁) → 𝑁 ≤ 𝑆))
62	61	adantld 490	. . . . . . . . . . . 12 ⊢ (𝜑 → ((¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁)) → 𝑁 ≤ 𝑆))
63	62	imp 406	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑁 ≤ 𝑆)
64		nesym 2989	. . . . . . . . . . . . . 14 ⊢ (𝑆 ≠ 𝑁 ↔ ¬ 𝑁 = 𝑆)
65	64	biimpri 228	. . . . . . . . . . . . 13 ⊢ (¬ 𝑁 = 𝑆 → 𝑆 ≠ 𝑁)
66	65	adantr 480	. . . . . . . . . . . 12 ⊢ ((¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁) → 𝑆 ≠ 𝑁)
67	66	ad2antll 730	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑆 ≠ 𝑁)
68	55, 58, 63, 67	leneltd 11291	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (¬ 𝑁 = 0 ∧ (¬ 𝑁 = 𝑆 ∧ ¬ 𝑆 < 𝑁))) → 𝑁 < 𝑆)
69	44, 68	sylbir 235	. . . . . . . . 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 2743	. . . . . . . . 9 ⊢ ((𝜑 ∧ 0 < 𝑁 ∧ 𝑁 < 𝑆) → ⦋𝑁 / 𝑛⦌𝐵 = 𝑌)
72	35, 53, 69, 71	syl3anc 1374	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → ⦋𝑁 / 𝑛⦌𝐵 = 𝑌)
73	22	ad3antrrr 731	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → 𝑌 ∈ 𝑉)
74	72, 73	eqeltrd 2837	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) ∧ ¬ 𝑆 < 𝑁) → ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉)
75	34, 74	ifclda 4503	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) → if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵) ∈ 𝑉)
76	29, 75	ifclda 4503	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑁 = 0) → if(𝑁 = 𝑆, ⦋𝑁 / 𝑛⦌𝐶, if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵)) ∈ 𝑉)
77	24, 76	ifclda 4503	. . . 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 2837	. . 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 6945	. . 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 585	. 2 ⊢ (𝜑 → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))
82	21	eqcomd 2743	. . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → ⦋𝑁 / 𝑛⦌𝐴 = 𝑌)
83	32, 72	ifeqda 4504	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑁 = 0) ∧ ¬ 𝑁 = 𝑆) → if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵) = 𝑌)
84	27, 83	ifeqda 4504	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑁 = 0) → if(𝑁 = 𝑆, ⦋𝑁 / 𝑛⦌𝐶, if(𝑆 < 𝑁, ⦋𝑁 / 𝑛⦌𝐷, ⦋𝑁 / 𝑛⦌𝐵)) = 𝑌)
85	82, 84	ifeqda 4504	. 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 2776	1 ⊢ (𝜑 → (𝐺‘𝑁) = 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  [wsbc 3729  ⦋csb 3838  ifcif 4467   class class class wbr 5086   ↦ cmpt 5167  ‘cfv 6492  ℝcr 11028  0cc0 11029   < clt 11170   ≤ cle 11171  ℕcn 12165  ℕ₀cn0 12428

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 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

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 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-n0 12429

This theorem is referenced by:  fvmptnn04ifa  22825  fvmptnn04ifb  22826  fvmptnn04ifc  22827  fvmptnn04ifd  22828