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Theorem padicabv 25319

Description: The p-adic absolute value (with arbitrary base) is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)

**Hypotheses**
Ref	Expression
qrng.q	⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
qabsabv.a	⊢ 𝐴 = (AbsVal‘𝑄)
padic.f	⊢ 𝐹 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑁↑(𝑃 pCnt 𝑥))))

**Assertion**
Ref	Expression
padicabv	⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝐹 ∈ 𝐴)

Distinct variable groups: 𝑥,𝐴 𝑥,𝑁 𝑥,𝑄 𝑥,𝑃 Allowed substitution hint: 𝐹(𝑥)

Proof of Theorem padicabv Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qabsabv.a	. . 3 ⊢ 𝐴 = (AbsVal‘𝑄)
2	1	a1i 11	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝐴 = (AbsVal‘𝑄))
3		qrng.q	. . . 4 ⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
4	3	qrngbas 25308	. . 3 ⊢ ℚ = (Base‘𝑄)
5	4	a1i 11	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → ℚ = (Base‘𝑄))
6		qex 11800	. . 3 ⊢ ℚ ∈ V
7		cnfldadd 19751	. . . 4 ⊢ + = (+_g‘ℂ_fld)
8	3, 7	ressplusg 15993	. . 3 ⊢ (ℚ ∈ V → + = (+_g‘𝑄))
9	6, 8	mp1i 13	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → + = (+_g‘𝑄))
10		cnfldmul 19752	. . . 4 ⊢ · = (._r‘ℂ_fld)
11	3, 10	ressmulr 16006	. . 3 ⊢ (ℚ ∈ V → · = (._r‘𝑄))
12	6, 11	mp1i 13	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → · = (._r‘𝑄))
13	3	qrng0 25310	. . 3 ⊢ 0 = (0_g‘𝑄)
14	13	a1i 11	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 0 = (0_g‘𝑄))
15	3	qdrng 25309	. . 3 ⊢ 𝑄 ∈ DivRing
16		drngring 18754	. . 3 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ Ring)
17	15, 16	mp1i 13	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝑄 ∈ Ring)
18		0red 10041	. . . 4 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑥 ∈ ℚ) ∧ 𝑥 = 0) → 0 ∈ ℝ)
19		ioossre 12235	. . . . . . 7 ⊢ (0(,)1) ⊆ ℝ
20		simpr 477	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝑁 ∈ (0(,)1))
21	19, 20	sseldi 3601	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝑁 ∈ ℝ)
22	21	ad2antrr 762	. . . . 5 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → 𝑁 ∈ ℝ)
23		eliooord 12233	. . . . . . . . . 10 ⊢ (𝑁 ∈ (0(,)1) → (0 < 𝑁 ∧ 𝑁 < 1))
24	23	adantl 482	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → (0 < 𝑁 ∧ 𝑁 < 1))
25	24	simpld 475	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 0 < 𝑁)
26	21, 25	elrpd 11869	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝑁 ∈ ℝ⁺)
27	26	rpne0d 11877	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝑁 ≠ 0)
28	27	ad2antrr 762	. . . . 5 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → 𝑁 ≠ 0)
29		df-ne 2795	. . . . . 6 ⊢ (𝑥 ≠ 0 ↔ ¬ 𝑥 = 0)
30		pcqcl 15561	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℚ ∧ 𝑥 ≠ 0)) → (𝑃 pCnt 𝑥) ∈ ℤ)
31	30	adantlr 751	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑥 ∈ ℚ ∧ 𝑥 ≠ 0)) → (𝑃 pCnt 𝑥) ∈ ℤ)
32	31	anassrs 680	. . . . . 6 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑥 ∈ ℚ) ∧ 𝑥 ≠ 0) → (𝑃 pCnt 𝑥) ∈ ℤ)
33	29, 32	sylan2br 493	. . . . 5 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → (𝑃 pCnt 𝑥) ∈ ℤ)
34	22, 28, 33	reexpclzd 13034	. . . 4 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → (𝑁↑(𝑃 pCnt 𝑥)) ∈ ℝ)
35	18, 34	ifclda 4120	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑥 ∈ ℚ) → if(𝑥 = 0, 0, (𝑁↑(𝑃 pCnt 𝑥))) ∈ ℝ)
36		padic.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑁↑(𝑃 pCnt 𝑥))))
37	35, 36	fmptd 6385	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝐹:ℚ⟶ℝ)
38		0z 11388	. . . 4 ⊢ 0 ∈ ℤ
39		zq 11794	. . . 4 ⊢ (0 ∈ ℤ → 0 ∈ ℚ)
40	38, 39	ax-mp 5	. . 3 ⊢ 0 ∈ ℚ
41		iftrue 4092	. . . 4 ⊢ (𝑥 = 0 → if(𝑥 = 0, 0, (𝑁↑(𝑃 pCnt 𝑥))) = 0)
42		c0ex 10034	. . . 4 ⊢ 0 ∈ V
43	41, 36, 42	fvmpt 6282	. . 3 ⊢ (0 ∈ ℚ → (𝐹‘0) = 0)
44	40, 43	mp1i 13	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → (𝐹‘0) = 0)
45	21	3ad2ant1 1082	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → 𝑁 ∈ ℝ)
46		pcqcl 15561	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0)) → (𝑃 pCnt 𝑦) ∈ ℤ)
47	46	adantlr 751	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0)) → (𝑃 pCnt 𝑦) ∈ ℤ)
48	47	3impb 1260	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → (𝑃 pCnt 𝑦) ∈ ℤ)
49	25	3ad2ant1 1082	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → 0 < 𝑁)
50		expgt0 12893	. . . 4 ⊢ ((𝑁 ∈ ℝ ∧ (𝑃 pCnt 𝑦) ∈ ℤ ∧ 0 < 𝑁) → 0 < (𝑁↑(𝑃 pCnt 𝑦)))
51	45, 48, 49, 50	syl3anc 1326	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → 0 < (𝑁↑(𝑃 pCnt 𝑦)))
52		eqeq1 2626	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 0 ↔ 𝑦 = 0))
53		oveq2 6658	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑃 pCnt 𝑥) = (𝑃 pCnt 𝑦))
54	53	oveq2d 6666	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑁↑(𝑃 pCnt 𝑥)) = (𝑁↑(𝑃 pCnt 𝑦)))
55	52, 54	ifbieq2d 4111	. . . . . 6 ⊢ (𝑥 = 𝑦 → if(𝑥 = 0, 0, (𝑁↑(𝑃 pCnt 𝑥))) = if(𝑦 = 0, 0, (𝑁↑(𝑃 pCnt 𝑦))))
56		ovex 6678	. . . . . . 7 ⊢ (𝑁↑(𝑃 pCnt 𝑦)) ∈ V
57	42, 56	ifex 4156	. . . . . 6 ⊢ if(𝑦 = 0, 0, (𝑁↑(𝑃 pCnt 𝑦))) ∈ V
58	55, 36, 57	fvmpt 6282	. . . . 5 ⊢ (𝑦 ∈ ℚ → (𝐹‘𝑦) = if(𝑦 = 0, 0, (𝑁↑(𝑃 pCnt 𝑦))))
59	58	3ad2ant2 1083	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → (𝐹‘𝑦) = if(𝑦 = 0, 0, (𝑁↑(𝑃 pCnt 𝑦))))
60		simp3 1063	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → 𝑦 ≠ 0)
61	60	neneqd 2799	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → ¬ 𝑦 = 0)
62	61	iffalsed 4097	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → if(𝑦 = 0, 0, (𝑁↑(𝑃 pCnt 𝑦))) = (𝑁↑(𝑃 pCnt 𝑦)))
63	59, 62	eqtrd 2656	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → (𝐹‘𝑦) = (𝑁↑(𝑃 pCnt 𝑦)))
64	51, 63	breqtrrd 4681	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → 0 < (𝐹‘𝑦))
65		pcqmul 15558	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑃 pCnt (𝑦 · 𝑧)) = ((𝑃 pCnt 𝑦) + (𝑃 pCnt 𝑧)))
66	65	3adant1r 1319	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑃 pCnt (𝑦 · 𝑧)) = ((𝑃 pCnt 𝑦) + (𝑃 pCnt 𝑧)))
67	66	oveq2d 6666	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑁↑(𝑃 pCnt (𝑦 · 𝑧))) = (𝑁↑((𝑃 pCnt 𝑦) + (𝑃 pCnt 𝑧))))
68	21	recnd 10068	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝑁 ∈ ℂ)
69	68	3ad2ant1 1082	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 𝑁 ∈ ℂ)
70	27	3ad2ant1 1082	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 𝑁 ≠ 0)
71	47	3adant3 1081	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑃 pCnt 𝑦) ∈ ℤ)
72		simp1l 1085	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 𝑃 ∈ ℙ)
73		simp3l 1089	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 𝑧 ∈ ℚ)
74		simp3r 1090	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 𝑧 ≠ 0)
75		pcqcl 15561	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑃 pCnt 𝑧) ∈ ℤ)
76	72, 73, 74, 75	syl12anc 1324	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑃 pCnt 𝑧) ∈ ℤ)
77		expaddz 12904	. . . . 5 ⊢ (((𝑁 ∈ ℂ ∧ 𝑁 ≠ 0) ∧ ((𝑃 pCnt 𝑦) ∈ ℤ ∧ (𝑃 pCnt 𝑧) ∈ ℤ)) → (𝑁↑((𝑃 pCnt 𝑦) + (𝑃 pCnt 𝑧))) = ((𝑁↑(𝑃 pCnt 𝑦)) · (𝑁↑(𝑃 pCnt 𝑧))))
78	69, 70, 71, 76, 77	syl22anc 1327	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑁↑((𝑃 pCnt 𝑦) + (𝑃 pCnt 𝑧))) = ((𝑁↑(𝑃 pCnt 𝑦)) · (𝑁↑(𝑃 pCnt 𝑧))))
79	67, 78	eqtrd 2656	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑁↑(𝑃 pCnt (𝑦 · 𝑧))) = ((𝑁↑(𝑃 pCnt 𝑦)) · (𝑁↑(𝑃 pCnt 𝑧))))
80		simp2l 1087	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 𝑦 ∈ ℚ)
81		qmulcl 11806	. . . . . 6 ⊢ ((𝑦 ∈ ℚ ∧ 𝑧 ∈ ℚ) → (𝑦 · 𝑧) ∈ ℚ)
82	80, 73, 81	syl2anc 693	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑦 · 𝑧) ∈ ℚ)
83		eqeq1 2626	. . . . . . 7 ⊢ (𝑥 = (𝑦 · 𝑧) → (𝑥 = 0 ↔ (𝑦 · 𝑧) = 0))
84		oveq2 6658	. . . . . . . 8 ⊢ (𝑥 = (𝑦 · 𝑧) → (𝑃 pCnt 𝑥) = (𝑃 pCnt (𝑦 · 𝑧)))
85	84	oveq2d 6666	. . . . . . 7 ⊢ (𝑥 = (𝑦 · 𝑧) → (𝑁↑(𝑃 pCnt 𝑥)) = (𝑁↑(𝑃 pCnt (𝑦 · 𝑧))))
86	83, 85	ifbieq2d 4111	. . . . . 6 ⊢ (𝑥 = (𝑦 · 𝑧) → if(𝑥 = 0, 0, (𝑁↑(𝑃 pCnt 𝑥))) = if((𝑦 · 𝑧) = 0, 0, (𝑁↑(𝑃 pCnt (𝑦 · 𝑧)))))
87		ovex 6678	. . . . . . 7 ⊢ (𝑁↑(𝑃 pCnt (𝑦 · 𝑧))) ∈ V
88	42, 87	ifex 4156	. . . . . 6 ⊢ if((𝑦 · 𝑧) = 0, 0, (𝑁↑(𝑃 pCnt (𝑦 · 𝑧)))) ∈ V
89	86, 36, 88	fvmpt 6282	. . . . 5 ⊢ ((𝑦 · 𝑧) ∈ ℚ → (𝐹‘(𝑦 · 𝑧)) = if((𝑦 · 𝑧) = 0, 0, (𝑁↑(𝑃 pCnt (𝑦 · 𝑧)))))
90	82, 89	syl 17	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝐹‘(𝑦 · 𝑧)) = if((𝑦 · 𝑧) = 0, 0, (𝑁↑(𝑃 pCnt (𝑦 · 𝑧)))))
91		qcn 11802	. . . . . . . 8 ⊢ (𝑦 ∈ ℚ → 𝑦 ∈ ℂ)
92	80, 91	syl 17	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 𝑦 ∈ ℂ)
93		qcn 11802	. . . . . . . 8 ⊢ (𝑧 ∈ ℚ → 𝑧 ∈ ℂ)
94	73, 93	syl 17	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 𝑧 ∈ ℂ)
95		simp2r 1088	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 𝑦 ≠ 0)
96	92, 94, 95, 74	mulne0d 10679	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑦 · 𝑧) ≠ 0)
97	96	neneqd 2799	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → ¬ (𝑦 · 𝑧) = 0)
98	97	iffalsed 4097	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → if((𝑦 · 𝑧) = 0, 0, (𝑁↑(𝑃 pCnt (𝑦 · 𝑧)))) = (𝑁↑(𝑃 pCnt (𝑦 · 𝑧))))
99	90, 98	eqtrd 2656	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝐹‘(𝑦 · 𝑧)) = (𝑁↑(𝑃 pCnt (𝑦 · 𝑧))))
100	63	3expb 1266	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0)) → (𝐹‘𝑦) = (𝑁↑(𝑃 pCnt 𝑦)))
101	100	3adant3 1081	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝐹‘𝑦) = (𝑁↑(𝑃 pCnt 𝑦)))
102		eqeq1 2626	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = 0 ↔ 𝑧 = 0))
103		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑃 pCnt 𝑥) = (𝑃 pCnt 𝑧))
104	103	oveq2d 6666	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑁↑(𝑃 pCnt 𝑥)) = (𝑁↑(𝑃 pCnt 𝑧)))
105	102, 104	ifbieq2d 4111	. . . . . . 7 ⊢ (𝑥 = 𝑧 → if(𝑥 = 0, 0, (𝑁↑(𝑃 pCnt 𝑥))) = if(𝑧 = 0, 0, (𝑁↑(𝑃 pCnt 𝑧))))
106		ovex 6678	. . . . . . . 8 ⊢ (𝑁↑(𝑃 pCnt 𝑧)) ∈ V
107	42, 106	ifex 4156	. . . . . . 7 ⊢ if(𝑧 = 0, 0, (𝑁↑(𝑃 pCnt 𝑧))) ∈ V
108	105, 36, 107	fvmpt 6282	. . . . . 6 ⊢ (𝑧 ∈ ℚ → (𝐹‘𝑧) = if(𝑧 = 0, 0, (𝑁↑(𝑃 pCnt 𝑧))))
109	73, 108	syl 17	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝐹‘𝑧) = if(𝑧 = 0, 0, (𝑁↑(𝑃 pCnt 𝑧))))
110	74	neneqd 2799	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → ¬ 𝑧 = 0)
111	110	iffalsed 4097	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → if(𝑧 = 0, 0, (𝑁↑(𝑃 pCnt 𝑧))) = (𝑁↑(𝑃 pCnt 𝑧)))
112	109, 111	eqtrd 2656	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝐹‘𝑧) = (𝑁↑(𝑃 pCnt 𝑧)))
113	101, 112	oveq12d 6668	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → ((𝐹‘𝑦) · (𝐹‘𝑧)) = ((𝑁↑(𝑃 pCnt 𝑦)) · (𝑁↑(𝑃 pCnt 𝑧))))
114	79, 99, 113	3eqtr4d 2666	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝐹‘(𝑦 · 𝑧)) = ((𝐹‘𝑦) · (𝐹‘𝑧)))
115		iftrue 4092	. . . . 5 ⊢ ((𝑦 + 𝑧) = 0 → if((𝑦 + 𝑧) = 0, 0, (𝑁↑(𝑃 pCnt (𝑦 + 𝑧)))) = 0)
116	115	breq1d 4663	. . . 4 ⊢ ((𝑦 + 𝑧) = 0 → (if((𝑦 + 𝑧) = 0, 0, (𝑁↑(𝑃 pCnt (𝑦 + 𝑧)))) ≤ ((𝑁↑(𝑃 pCnt 𝑦)) + (𝑁↑(𝑃 pCnt 𝑧))) ↔ 0 ≤ ((𝑁↑(𝑃 pCnt 𝑦)) + (𝑁↑(𝑃 pCnt 𝑧)))))
117		ifnefalse 4098	. . . . . 6 ⊢ ((𝑦 + 𝑧) ≠ 0 → if((𝑦 + 𝑧) = 0, 0, (𝑁↑(𝑃 pCnt (𝑦 + 𝑧)))) = (𝑁↑(𝑃 pCnt (𝑦 + 𝑧))))
118	117	adantl 482	. . . . 5 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → if((𝑦 + 𝑧) = 0, 0, (𝑁↑(𝑃 pCnt (𝑦 + 𝑧)))) = (𝑁↑(𝑃 pCnt (𝑦 + 𝑧))))
119	71	adantr 481	. . . . . . 7 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (𝑃 pCnt 𝑦) ∈ ℤ)
120	119	zred 11482	. . . . . 6 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (𝑃 pCnt 𝑦) ∈ ℝ)
121	76	adantr 481	. . . . . . 7 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (𝑃 pCnt 𝑧) ∈ ℤ)
122	121	zred 11482	. . . . . 6 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (𝑃 pCnt 𝑧) ∈ ℝ)
123	21	3ad2ant1 1082	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 𝑁 ∈ ℝ)
124	123	ad2antrr 762	. . . . . . . 8 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑦) ≤ (𝑃 pCnt 𝑧)) → 𝑁 ∈ ℝ)
125	70	ad2antrr 762	. . . . . . . 8 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑦) ≤ (𝑃 pCnt 𝑧)) → 𝑁 ≠ 0)
126	72	adantr 481	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → 𝑃 ∈ ℙ)
127		qaddcl 11804	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℚ ∧ 𝑧 ∈ ℚ) → (𝑦 + 𝑧) ∈ ℚ)
128	80, 73, 127	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑦 + 𝑧) ∈ ℚ)
129	128	adantr 481	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (𝑦 + 𝑧) ∈ ℚ)
130		simpr 477	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (𝑦 + 𝑧) ≠ 0)
131		pcqcl 15561	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ ((𝑦 + 𝑧) ∈ ℚ ∧ (𝑦 + 𝑧) ≠ 0)) → (𝑃 pCnt (𝑦 + 𝑧)) ∈ ℤ)
132	126, 129, 130, 131	syl12anc 1324	. . . . . . . . 9 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (𝑃 pCnt (𝑦 + 𝑧)) ∈ ℤ)
133	132	adantr 481	. . . . . . . 8 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑦) ≤ (𝑃 pCnt 𝑧)) → (𝑃 pCnt (𝑦 + 𝑧)) ∈ ℤ)
134	124, 125, 133	reexpclzd 13034	. . . . . . 7 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑦) ≤ (𝑃 pCnt 𝑧)) → (𝑁↑(𝑃 pCnt (𝑦 + 𝑧))) ∈ ℝ)
135	119	adantr 481	. . . . . . . 8 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑦) ≤ (𝑃 pCnt 𝑧)) → (𝑃 pCnt 𝑦) ∈ ℤ)
136	124, 125, 135	reexpclzd 13034	. . . . . . 7 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑦) ≤ (𝑃 pCnt 𝑧)) → (𝑁↑(𝑃 pCnt 𝑦)) ∈ ℝ)
137		simpl1 1064	. . . . . . . . . . 11 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)))
138	137, 21	syl 17	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → 𝑁 ∈ ℝ)
139	137, 27	syl 17	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → 𝑁 ≠ 0)
140	138, 139, 119	reexpclzd 13034	. . . . . . . . 9 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (𝑁↑(𝑃 pCnt 𝑦)) ∈ ℝ)
141	138, 139, 121	reexpclzd 13034	. . . . . . . . 9 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (𝑁↑(𝑃 pCnt 𝑧)) ∈ ℝ)
142	140, 141	readdcld 10069	. . . . . . . 8 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → ((𝑁↑(𝑃 pCnt 𝑦)) + (𝑁↑(𝑃 pCnt 𝑧))) ∈ ℝ)
143	142	adantr 481	. . . . . . 7 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑦) ≤ (𝑃 pCnt 𝑧)) → ((𝑁↑(𝑃 pCnt 𝑦)) + (𝑁↑(𝑃 pCnt 𝑧))) ∈ ℝ)
144	126	adantr 481	. . . . . . . . 9 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑦) ≤ (𝑃 pCnt 𝑧)) → 𝑃 ∈ ℙ)
145	80	ad2antrr 762	. . . . . . . . 9 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑦) ≤ (𝑃 pCnt 𝑧)) → 𝑦 ∈ ℚ)
146	73	ad2antrr 762	. . . . . . . . 9 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑦) ≤ (𝑃 pCnt 𝑧)) → 𝑧 ∈ ℚ)
147		simpr 477	. . . . . . . . 9 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑦) ≤ (𝑃 pCnt 𝑧)) → (𝑃 pCnt 𝑦) ≤ (𝑃 pCnt 𝑧))
148	144, 145, 146, 147	pcadd 15593	. . . . . . . 8 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑦) ≤ (𝑃 pCnt 𝑧)) → (𝑃 pCnt 𝑦) ≤ (𝑃 pCnt (𝑦 + 𝑧)))
149	137, 26	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → 𝑁 ∈ ℝ⁺)
150	24	simprd 479	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝑁 < 1)
151	137, 150	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → 𝑁 < 1)
152	149, 119, 132, 151	ltexp2rd 13033	. . . . . . . . . . 11 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → ((𝑃 pCnt (𝑦 + 𝑧)) < (𝑃 pCnt 𝑦) ↔ (𝑁↑(𝑃 pCnt 𝑦)) < (𝑁↑(𝑃 pCnt (𝑦 + 𝑧)))))
153	152	notbid 308	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (¬ (𝑃 pCnt (𝑦 + 𝑧)) < (𝑃 pCnt 𝑦) ↔ ¬ (𝑁↑(𝑃 pCnt 𝑦)) < (𝑁↑(𝑃 pCnt (𝑦 + 𝑧)))))
154	132	zred 11482	. . . . . . . . . . 11 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (𝑃 pCnt (𝑦 + 𝑧)) ∈ ℝ)
155	120, 154	lenltd 10183	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → ((𝑃 pCnt 𝑦) ≤ (𝑃 pCnt (𝑦 + 𝑧)) ↔ ¬ (𝑃 pCnt (𝑦 + 𝑧)) < (𝑃 pCnt 𝑦)))
156	138, 139, 132	reexpclzd 13034	. . . . . . . . . . 11 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (𝑁↑(𝑃 pCnt (𝑦 + 𝑧))) ∈ ℝ)
157	156, 140	lenltd 10183	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → ((𝑁↑(𝑃 pCnt (𝑦 + 𝑧))) ≤ (𝑁↑(𝑃 pCnt 𝑦)) ↔ ¬ (𝑁↑(𝑃 pCnt 𝑦)) < (𝑁↑(𝑃 pCnt (𝑦 + 𝑧)))))
158	153, 155, 157	3bitr4d 300	. . . . . . . . 9 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → ((𝑃 pCnt 𝑦) ≤ (𝑃 pCnt (𝑦 + 𝑧)) ↔ (𝑁↑(𝑃 pCnt (𝑦 + 𝑧))) ≤ (𝑁↑(𝑃 pCnt 𝑦))))
159	158	biimpa 501	. . . . . . . 8 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑦) ≤ (𝑃 pCnt (𝑦 + 𝑧))) → (𝑁↑(𝑃 pCnt (𝑦 + 𝑧))) ≤ (𝑁↑(𝑃 pCnt 𝑦)))
160	148, 159	syldan 487	. . . . . . 7 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑦) ≤ (𝑃 pCnt 𝑧)) → (𝑁↑(𝑃 pCnt (𝑦 + 𝑧))) ≤ (𝑁↑(𝑃 pCnt 𝑦)))
161	26	3ad2ant1 1082	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 𝑁 ∈ ℝ⁺)
162	161, 76	rpexpcld 13032	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑁↑(𝑃 pCnt 𝑧)) ∈ ℝ⁺)
163	162	adantr 481	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (𝑁↑(𝑃 pCnt 𝑧)) ∈ ℝ⁺)
164	163	rpge0d 11876	. . . . . . . . 9 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → 0 ≤ (𝑁↑(𝑃 pCnt 𝑧)))
165	140, 141	addge01d 10615	. . . . . . . . 9 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (0 ≤ (𝑁↑(𝑃 pCnt 𝑧)) ↔ (𝑁↑(𝑃 pCnt 𝑦)) ≤ ((𝑁↑(𝑃 pCnt 𝑦)) + (𝑁↑(𝑃 pCnt 𝑧)))))
166	164, 165	mpbid 222	. . . . . . . 8 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (𝑁↑(𝑃 pCnt 𝑦)) ≤ ((𝑁↑(𝑃 pCnt 𝑦)) + (𝑁↑(𝑃 pCnt 𝑧))))
167	166	adantr 481	. . . . . . 7 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑦) ≤ (𝑃 pCnt 𝑧)) → (𝑁↑(𝑃 pCnt 𝑦)) ≤ ((𝑁↑(𝑃 pCnt 𝑦)) + (𝑁↑(𝑃 pCnt 𝑧))))
168	134, 136, 143, 160, 167	letrd 10194	. . . . . 6 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑦) ≤ (𝑃 pCnt 𝑧)) → (𝑁↑(𝑃 pCnt (𝑦 + 𝑧))) ≤ ((𝑁↑(𝑃 pCnt 𝑦)) + (𝑁↑(𝑃 pCnt 𝑧))))
169	156	adantr 481	. . . . . . 7 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑧) ≤ (𝑃 pCnt 𝑦)) → (𝑁↑(𝑃 pCnt (𝑦 + 𝑧))) ∈ ℝ)
170	141	adantr 481	. . . . . . 7 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑧) ≤ (𝑃 pCnt 𝑦)) → (𝑁↑(𝑃 pCnt 𝑧)) ∈ ℝ)
171	142	adantr 481	. . . . . . 7 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑧) ≤ (𝑃 pCnt 𝑦)) → ((𝑁↑(𝑃 pCnt 𝑦)) + (𝑁↑(𝑃 pCnt 𝑧))) ∈ ℝ)
172	126	adantr 481	. . . . . . . . . 10 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑧) ≤ (𝑃 pCnt 𝑦)) → 𝑃 ∈ ℙ)
173	73	ad2antrr 762	. . . . . . . . . 10 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑧) ≤ (𝑃 pCnt 𝑦)) → 𝑧 ∈ ℚ)
174	80	ad2antrr 762	. . . . . . . . . 10 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑧) ≤ (𝑃 pCnt 𝑦)) → 𝑦 ∈ ℚ)
175		simpr 477	. . . . . . . . . 10 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑧) ≤ (𝑃 pCnt 𝑦)) → (𝑃 pCnt 𝑧) ≤ (𝑃 pCnt 𝑦))
176	172, 173, 174, 175	pcadd 15593	. . . . . . . . 9 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑧) ≤ (𝑃 pCnt 𝑦)) → (𝑃 pCnt 𝑧) ≤ (𝑃 pCnt (𝑧 + 𝑦)))
177	92, 94	addcomd 10238	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑦 + 𝑧) = (𝑧 + 𝑦))
178	177	oveq2d 6666	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑃 pCnt (𝑦 + 𝑧)) = (𝑃 pCnt (𝑧 + 𝑦)))
179	178	ad2antrr 762	. . . . . . . . 9 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑧) ≤ (𝑃 pCnt 𝑦)) → (𝑃 pCnt (𝑦 + 𝑧)) = (𝑃 pCnt (𝑧 + 𝑦)))
180	176, 179	breqtrrd 4681	. . . . . . . 8 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑧) ≤ (𝑃 pCnt 𝑦)) → (𝑃 pCnt 𝑧) ≤ (𝑃 pCnt (𝑦 + 𝑧)))
181	149, 121, 132, 151	ltexp2rd 13033	. . . . . . . . . . 11 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → ((𝑃 pCnt (𝑦 + 𝑧)) < (𝑃 pCnt 𝑧) ↔ (𝑁↑(𝑃 pCnt 𝑧)) < (𝑁↑(𝑃 pCnt (𝑦 + 𝑧)))))
182	181	notbid 308	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (¬ (𝑃 pCnt (𝑦 + 𝑧)) < (𝑃 pCnt 𝑧) ↔ ¬ (𝑁↑(𝑃 pCnt 𝑧)) < (𝑁↑(𝑃 pCnt (𝑦 + 𝑧)))))
183	122, 154	lenltd 10183	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → ((𝑃 pCnt 𝑧) ≤ (𝑃 pCnt (𝑦 + 𝑧)) ↔ ¬ (𝑃 pCnt (𝑦 + 𝑧)) < (𝑃 pCnt 𝑧)))
184	156, 141	lenltd 10183	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → ((𝑁↑(𝑃 pCnt (𝑦 + 𝑧))) ≤ (𝑁↑(𝑃 pCnt 𝑧)) ↔ ¬ (𝑁↑(𝑃 pCnt 𝑧)) < (𝑁↑(𝑃 pCnt (𝑦 + 𝑧)))))
185	182, 183, 184	3bitr4d 300	. . . . . . . . 9 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → ((𝑃 pCnt 𝑧) ≤ (𝑃 pCnt (𝑦 + 𝑧)) ↔ (𝑁↑(𝑃 pCnt (𝑦 + 𝑧))) ≤ (𝑁↑(𝑃 pCnt 𝑧))))
186	185	biimpa 501	. . . . . . . 8 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑧) ≤ (𝑃 pCnt (𝑦 + 𝑧))) → (𝑁↑(𝑃 pCnt (𝑦 + 𝑧))) ≤ (𝑁↑(𝑃 pCnt 𝑧)))
187	180, 186	syldan 487	. . . . . . 7 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑧) ≤ (𝑃 pCnt 𝑦)) → (𝑁↑(𝑃 pCnt (𝑦 + 𝑧))) ≤ (𝑁↑(𝑃 pCnt 𝑧)))
188	161, 71	rpexpcld 13032	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑁↑(𝑃 pCnt 𝑦)) ∈ ℝ⁺)
189	188	adantr 481	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (𝑁↑(𝑃 pCnt 𝑦)) ∈ ℝ⁺)
190	189	rpge0d 11876	. . . . . . . . 9 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → 0 ≤ (𝑁↑(𝑃 pCnt 𝑦)))
191	141, 140	addge02d 10616	. . . . . . . . 9 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (0 ≤ (𝑁↑(𝑃 pCnt 𝑦)) ↔ (𝑁↑(𝑃 pCnt 𝑧)) ≤ ((𝑁↑(𝑃 pCnt 𝑦)) + (𝑁↑(𝑃 pCnt 𝑧)))))
192	190, 191	mpbid 222	. . . . . . . 8 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (𝑁↑(𝑃 pCnt 𝑧)) ≤ ((𝑁↑(𝑃 pCnt 𝑦)) + (𝑁↑(𝑃 pCnt 𝑧))))
193	192	adantr 481	. . . . . . 7 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑧) ≤ (𝑃 pCnt 𝑦)) → (𝑁↑(𝑃 pCnt 𝑧)) ≤ ((𝑁↑(𝑃 pCnt 𝑦)) + (𝑁↑(𝑃 pCnt 𝑧))))
194	169, 170, 171, 187, 193	letrd 10194	. . . . . 6 ⊢ (((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) ∧ (𝑃 pCnt 𝑧) ≤ (𝑃 pCnt 𝑦)) → (𝑁↑(𝑃 pCnt (𝑦 + 𝑧))) ≤ ((𝑁↑(𝑃 pCnt 𝑦)) + (𝑁↑(𝑃 pCnt 𝑧))))
195	120, 122, 168, 194	lecasei 10143	. . . . 5 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (𝑁↑(𝑃 pCnt (𝑦 + 𝑧))) ≤ ((𝑁↑(𝑃 pCnt 𝑦)) + (𝑁↑(𝑃 pCnt 𝑧))))
196	118, 195	eqbrtrd 4675	. . . 4 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → if((𝑦 + 𝑧) = 0, 0, (𝑁↑(𝑃 pCnt (𝑦 + 𝑧)))) ≤ ((𝑁↑(𝑃 pCnt 𝑦)) + (𝑁↑(𝑃 pCnt 𝑧))))
197	188, 162	rpaddcld 11887	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → ((𝑁↑(𝑃 pCnt 𝑦)) + (𝑁↑(𝑃 pCnt 𝑧))) ∈ ℝ⁺)
198	197	rpge0d 11876	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 0 ≤ ((𝑁↑(𝑃 pCnt 𝑦)) + (𝑁↑(𝑃 pCnt 𝑧))))
199	116, 196, 198	pm2.61ne 2879	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → if((𝑦 + 𝑧) = 0, 0, (𝑁↑(𝑃 pCnt (𝑦 + 𝑧)))) ≤ ((𝑁↑(𝑃 pCnt 𝑦)) + (𝑁↑(𝑃 pCnt 𝑧))))
200		eqeq1 2626	. . . . . 6 ⊢ (𝑥 = (𝑦 + 𝑧) → (𝑥 = 0 ↔ (𝑦 + 𝑧) = 0))
201		oveq2 6658	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 𝑧) → (𝑃 pCnt 𝑥) = (𝑃 pCnt (𝑦 + 𝑧)))
202	201	oveq2d 6666	. . . . . 6 ⊢ (𝑥 = (𝑦 + 𝑧) → (𝑁↑(𝑃 pCnt 𝑥)) = (𝑁↑(𝑃 pCnt (𝑦 + 𝑧))))
203	200, 202	ifbieq2d 4111	. . . . 5 ⊢ (𝑥 = (𝑦 + 𝑧) → if(𝑥 = 0, 0, (𝑁↑(𝑃 pCnt 𝑥))) = if((𝑦 + 𝑧) = 0, 0, (𝑁↑(𝑃 pCnt (𝑦 + 𝑧)))))
204		ovex 6678	. . . . . 6 ⊢ (𝑁↑(𝑃 pCnt (𝑦 + 𝑧))) ∈ V
205	42, 204	ifex 4156	. . . . 5 ⊢ if((𝑦 + 𝑧) = 0, 0, (𝑁↑(𝑃 pCnt (𝑦 + 𝑧)))) ∈ V
206	203, 36, 205	fvmpt 6282	. . . 4 ⊢ ((𝑦 + 𝑧) ∈ ℚ → (𝐹‘(𝑦 + 𝑧)) = if((𝑦 + 𝑧) = 0, 0, (𝑁↑(𝑃 pCnt (𝑦 + 𝑧)))))
207	128, 206	syl 17	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝐹‘(𝑦 + 𝑧)) = if((𝑦 + 𝑧) = 0, 0, (𝑁↑(𝑃 pCnt (𝑦 + 𝑧)))))
208	101, 112	oveq12d 6668	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → ((𝐹‘𝑦) + (𝐹‘𝑧)) = ((𝑁↑(𝑃 pCnt 𝑦)) + (𝑁↑(𝑃 pCnt 𝑧))))
209	199, 207, 208	3brtr4d 4685	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝐹‘(𝑦 + 𝑧)) ≤ ((𝐹‘𝑦) + (𝐹‘𝑧)))
210	2, 5, 9, 12, 14, 17, 37, 44, 64, 114, 209	isabvd 18820	1 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝐹 ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ifcif 4086 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 ≤ cle 10075 ℤcz 11377 ℚcq 11788 ℝ⁺crp 11832 (,)cioo 12175 ↑cexp 12860 ℙcprime 15385 pCnt cpc 15541 Basecbs 15857 ↾_s cress 15858 +_gcplusg 15941 ._rcmulr 15942 0_gc0g 16100 Ringcrg 18547 DivRingcdr 18747 AbsValcabv 18816 ℂ_fldccnfld 19746

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-ioo 12179 df-ico 12181 df-fz 12327 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-subg 17591 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-subrg 18778 df-abv 18817 df-cnfld 19747

This theorem is referenced by: padicabvf 25320 padicabvcxp 25321

W3C validator