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Theorem padicabv 26978

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 26967	. . 3 ⊢ ℚ = (Base‘𝑄)
5	4	a1i 11	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → ℚ = (Base‘𝑄))
6		qex 12886	. . 3 ⊢ ℚ ∈ V
7		cnfldadd 20801	. . . 4 ⊢ + = (+_g‘ℂ_fld)
8	3, 7	ressplusg 17171	. . 3 ⊢ (ℚ ∈ V → + = (+_g‘𝑄))
9	6, 8	mp1i 13	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → + = (+_g‘𝑄))
10		cnfldmul 20802	. . . 4 ⊢ · = (._r‘ℂ_fld)
11	3, 10	ressmulr 17188	. . 3 ⊢ (ℚ ∈ V → · = (._r‘𝑄))
12	6, 11	mp1i 13	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → · = (._r‘𝑄))
13	3	qrng0 26969	. . 3 ⊢ 0 = (0_g‘𝑄)
14	13	a1i 11	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 0 = (0_g‘𝑄))
15	3	qdrng 26968	. . 3 ⊢ 𝑄 ∈ DivRing
16		drngring 20192	. . 3 ⊢ (𝑄 ∈ DivRing → 𝑄 ∈ Ring)
17	15, 16	mp1i 13	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝑄 ∈ Ring)
18		0red 11158	. . . 4 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑥 ∈ ℚ) ∧ 𝑥 = 0) → 0 ∈ ℝ)
19		ioossre 13325	. . . . . . 7 ⊢ (0(,)1) ⊆ ℝ
20		simpr 485	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝑁 ∈ (0(,)1))
21	19, 20	sselid 3942	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝑁 ∈ ℝ)
22	21	ad2antrr 724	. . . . 5 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → 𝑁 ∈ ℝ)
23		eliooord 13323	. . . . . . . . . 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 495	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 0 < 𝑁)
26	21, 25	elrpd 12954	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝑁 ∈ ℝ⁺)
27	26	rpne0d 12962	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝑁 ≠ 0)
28	27	ad2antrr 724	. . . . 5 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → 𝑁 ≠ 0)
29		df-ne 2944	. . . . . 6 ⊢ (𝑥 ≠ 0 ↔ ¬ 𝑥 = 0)
30		pcqcl 16728	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℚ ∧ 𝑥 ≠ 0)) → (𝑃 pCnt 𝑥) ∈ ℤ)
31	30	adantlr 713	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑥 ∈ ℚ ∧ 𝑥 ≠ 0)) → (𝑃 pCnt 𝑥) ∈ ℤ)
32	31	anassrs 468	. . . . . 6 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑥 ∈ ℚ) ∧ 𝑥 ≠ 0) → (𝑃 pCnt 𝑥) ∈ ℤ)
33	29, 32	sylan2br 595	. . . . 5 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → (𝑃 pCnt 𝑥) ∈ ℤ)
34	22, 28, 33	reexpclzd 14152	. . . 4 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑥 ∈ ℚ) ∧ ¬ 𝑥 = 0) → (𝑁↑(𝑃 pCnt 𝑥)) ∈ ℝ)
35	18, 34	ifclda 4521	. . 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 7062	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝐹:ℚ⟶ℝ)
38		0z 12510	. . . 4 ⊢ 0 ∈ ℤ
39		zq 12879	. . . 4 ⊢ (0 ∈ ℤ → 0 ∈ ℚ)
40	38, 39	ax-mp 5	. . 3 ⊢ 0 ∈ ℚ
41		iftrue 4492	. . . 4 ⊢ (𝑥 = 0 → if(𝑥 = 0, 0, (𝑁↑(𝑃 pCnt 𝑥))) = 0)
42		c0ex 11149	. . . 4 ⊢ 0 ∈ V
43	41, 36, 42	fvmpt 6948	. . 3 ⊢ (0 ∈ ℚ → (𝐹‘0) = 0)
44	40, 43	mp1i 13	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → (𝐹‘0) = 0)
45	21	3ad2ant1 1133	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → 𝑁 ∈ ℝ)
46		pcqcl 16728	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0)) → (𝑃 pCnt 𝑦) ∈ ℤ)
47	46	adantlr 713	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0)) → (𝑃 pCnt 𝑦) ∈ ℤ)
48	47	3impb 1115	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → (𝑃 pCnt 𝑦) ∈ ℤ)
49	25	3ad2ant1 1133	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → 0 < 𝑁)
50		expgt0 14001	. . . 4 ⊢ ((𝑁 ∈ ℝ ∧ (𝑃 pCnt 𝑦) ∈ ℤ ∧ 0 < 𝑁) → 0 < (𝑁↑(𝑃 pCnt 𝑦)))
51	45, 48, 49, 50	syl3anc 1371	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → 0 < (𝑁↑(𝑃 pCnt 𝑦)))
52		eqeq1 2740	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 0 ↔ 𝑦 = 0))
53		oveq2 7365	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑃 pCnt 𝑥) = (𝑃 pCnt 𝑦))
54	53	oveq2d 7373	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑁↑(𝑃 pCnt 𝑥)) = (𝑁↑(𝑃 pCnt 𝑦)))
55	52, 54	ifbieq2d 4512	. . . . . 6 ⊢ (𝑥 = 𝑦 → if(𝑥 = 0, 0, (𝑁↑(𝑃 pCnt 𝑥))) = if(𝑦 = 0, 0, (𝑁↑(𝑃 pCnt 𝑦))))
56		ovex 7390	. . . . . . 7 ⊢ (𝑁↑(𝑃 pCnt 𝑦)) ∈ V
57	42, 56	ifex 4536	. . . . . 6 ⊢ if(𝑦 = 0, 0, (𝑁↑(𝑃 pCnt 𝑦))) ∈ V
58	55, 36, 57	fvmpt 6948	. . . . 5 ⊢ (𝑦 ∈ ℚ → (𝐹‘𝑦) = if(𝑦 = 0, 0, (𝑁↑(𝑃 pCnt 𝑦))))
59	58	3ad2ant2 1134	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → (𝐹‘𝑦) = if(𝑦 = 0, 0, (𝑁↑(𝑃 pCnt 𝑦))))
60		simp3 1138	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → 𝑦 ≠ 0)
61	60	neneqd 2948	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → ¬ 𝑦 = 0)
62	61	iffalsed 4497	. . . 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 2776	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → (𝐹‘𝑦) = (𝑁↑(𝑃 pCnt 𝑦)))
64	51, 63	breqtrrd 5133	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → 0 < (𝐹‘𝑦))
65		pcqmul 16725	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑃 pCnt (𝑦 · 𝑧)) = ((𝑃 pCnt 𝑦) + (𝑃 pCnt 𝑧)))
66	65	3adant1r 1177	. . . . 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 7373	. . . 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 11183	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝑁 ∈ ℂ)
69	68	3ad2ant1 1133	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 𝑁 ∈ ℂ)
70	27	3ad2ant1 1133	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 𝑁 ≠ 0)
71	47	3adant3 1132	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑃 pCnt 𝑦) ∈ ℤ)
72		simp1l 1197	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 𝑃 ∈ ℙ)
73		simp3l 1201	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 𝑧 ∈ ℚ)
74		simp3r 1202	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 𝑧 ≠ 0)
75		pcqcl 16728	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑃 pCnt 𝑧) ∈ ℤ)
76	72, 73, 74, 75	syl12anc 835	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑃 pCnt 𝑧) ∈ ℤ)
77		expaddz 14012	. . . . 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 837	. . . 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 2776	. . 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 1199	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 𝑦 ∈ ℚ)
81		qmulcl 12892	. . . . . 6 ⊢ ((𝑦 ∈ ℚ ∧ 𝑧 ∈ ℚ) → (𝑦 · 𝑧) ∈ ℚ)
82	80, 73, 81	syl2anc 584	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑦 · 𝑧) ∈ ℚ)
83		eqeq1 2740	. . . . . . 7 ⊢ (𝑥 = (𝑦 · 𝑧) → (𝑥 = 0 ↔ (𝑦 · 𝑧) = 0))
84		oveq2 7365	. . . . . . . 8 ⊢ (𝑥 = (𝑦 · 𝑧) → (𝑃 pCnt 𝑥) = (𝑃 pCnt (𝑦 · 𝑧)))
85	84	oveq2d 7373	. . . . . . 7 ⊢ (𝑥 = (𝑦 · 𝑧) → (𝑁↑(𝑃 pCnt 𝑥)) = (𝑁↑(𝑃 pCnt (𝑦 · 𝑧))))
86	83, 85	ifbieq2d 4512	. . . . . 6 ⊢ (𝑥 = (𝑦 · 𝑧) → if(𝑥 = 0, 0, (𝑁↑(𝑃 pCnt 𝑥))) = if((𝑦 · 𝑧) = 0, 0, (𝑁↑(𝑃 pCnt (𝑦 · 𝑧)))))
87		ovex 7390	. . . . . . 7 ⊢ (𝑁↑(𝑃 pCnt (𝑦 · 𝑧))) ∈ V
88	42, 87	ifex 4536	. . . . . 6 ⊢ if((𝑦 · 𝑧) = 0, 0, (𝑁↑(𝑃 pCnt (𝑦 · 𝑧)))) ∈ V
89	86, 36, 88	fvmpt 6948	. . . . 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 12888	. . . . . . . 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 12888	. . . . . . . 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 1200	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 𝑦 ≠ 0)
96	92, 94, 95, 74	mulne0d 11807	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑦 · 𝑧) ≠ 0)
97	96	neneqd 2948	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → ¬ (𝑦 · 𝑧) = 0)
98	97	iffalsed 4497	. . . 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 2776	. . 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 1120	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0)) → (𝐹‘𝑦) = (𝑁↑(𝑃 pCnt 𝑦)))
101	100	3adant3 1132	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝐹‘𝑦) = (𝑁↑(𝑃 pCnt 𝑦)))
102		eqeq1 2740	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = 0 ↔ 𝑧 = 0))
103		oveq2 7365	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑃 pCnt 𝑥) = (𝑃 pCnt 𝑧))
104	103	oveq2d 7373	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑁↑(𝑃 pCnt 𝑥)) = (𝑁↑(𝑃 pCnt 𝑧)))
105	102, 104	ifbieq2d 4512	. . . . . . 7 ⊢ (𝑥 = 𝑧 → if(𝑥 = 0, 0, (𝑁↑(𝑃 pCnt 𝑥))) = if(𝑧 = 0, 0, (𝑁↑(𝑃 pCnt 𝑧))))
106		ovex 7390	. . . . . . . 8 ⊢ (𝑁↑(𝑃 pCnt 𝑧)) ∈ V
107	42, 106	ifex 4536	. . . . . . 7 ⊢ if(𝑧 = 0, 0, (𝑁↑(𝑃 pCnt 𝑧))) ∈ V
108	105, 36, 107	fvmpt 6948	. . . . . 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 2948	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → ¬ 𝑧 = 0)
111	110	iffalsed 4497	. . . . 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 2776	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝐹‘𝑧) = (𝑁↑(𝑃 pCnt 𝑧)))
113	101, 112	oveq12d 7375	. . 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 2786	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝐹‘(𝑦 · 𝑧)) = ((𝐹‘𝑦) · (𝐹‘𝑧)))
115		iftrue 4492	. . . . 5 ⊢ ((𝑦 + 𝑧) = 0 → if((𝑦 + 𝑧) = 0, 0, (𝑁↑(𝑃 pCnt (𝑦 + 𝑧)))) = 0)
116	115	breq1d 5115	. . . 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 4498	. . . . . 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 12607	. . . . . 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 12607	. . . . . 6 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (𝑃 pCnt 𝑧) ∈ ℝ)
123	21	3ad2ant1 1133	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 𝑁 ∈ ℝ)
124	123	ad2antrr 724	. . . . . . . 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 724	. . . . . . . 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 12890	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℚ ∧ 𝑧 ∈ ℚ) → (𝑦 + 𝑧) ∈ ℚ)
128	80, 73, 127	syl2anc 584	. . . . . . . . . . 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 485	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) ∧ (𝑦 + 𝑧) ≠ 0) → (𝑦 + 𝑧) ≠ 0)
131		pcqcl 16728	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ ((𝑦 + 𝑧) ∈ ℚ ∧ (𝑦 + 𝑧) ≠ 0)) → (𝑃 pCnt (𝑦 + 𝑧)) ∈ ℤ)
132	126, 129, 130, 131	syl12anc 835	. . . . . . . . 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 14152	. . . . . . 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 14152	. . . . . . 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 1191	. . . . . . . . . . 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 14152	. . . . . . . . 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 14152	. . . . . . . . 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 11184	. . . . . . . 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 724	. . . . . . . . 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 724	. . . . . . . . 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 485	. . . . . . . . 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 16761	. . . . . . . 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 496	. . . . . . . . . . . . 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 14151	. . . . . . . . . . 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 317	. . . . . . . . . 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 12607	. . . . . . . . . . 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 11301	. . . . . . . . . 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 14152	. . . . . . . . . . 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 11301	. . . . . . . . . 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 310	. . . . . . . . 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 477	. . . . . . . 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 591	. . . . . . 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 1133	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → 𝑁 ∈ ℝ⁺)
162	161, 76	rpexpcld 14150	. . . . . . . . . . 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 12961	. . . . . . . . 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 11743	. . . . . . . . 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 231	. . . . . . . 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 11312	. . . . . 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 724	. . . . . . . . . 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 724	. . . . . . . . . 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 485	. . . . . . . . . 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 16761	. . . . . . . . 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 11357	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) ∧ (𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) ∧ (𝑧 ∈ ℚ ∧ 𝑧 ≠ 0)) → (𝑦 + 𝑧) = (𝑧 + 𝑦))
178	177	oveq2d 7373	. . . . . . . . . 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 724	. . . . . . . . 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 5133	. . . . . . . 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 14151	. . . . . . . . . . 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 317	. . . . . . . . . 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 11301	. . . . . . . . . 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 11301	. . . . . . . . . 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 310	. . . . . . . . 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 477	. . . . . . . 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 591	. . . . . . 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 14150	. . . . . . . . . . 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 12961	. . . . . . . . 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 11744	. . . . . . . . 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 231	. . . . . . . 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 11312	. . . . . 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 11261	. . . . 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 5127	. . . 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 12972	. . . . 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 12961	. . . 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 3030	. . 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 2740	. . . . . 6 ⊢ (𝑥 = (𝑦 + 𝑧) → (𝑥 = 0 ↔ (𝑦 + 𝑧) = 0))
201		oveq2 7365	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 𝑧) → (𝑃 pCnt 𝑥) = (𝑃 pCnt (𝑦 + 𝑧)))
202	201	oveq2d 7373	. . . . . 6 ⊢ (𝑥 = (𝑦 + 𝑧) → (𝑁↑(𝑃 pCnt 𝑥)) = (𝑁↑(𝑃 pCnt (𝑦 + 𝑧))))
203	200, 202	ifbieq2d 4512	. . . . 5 ⊢ (𝑥 = (𝑦 + 𝑧) → if(𝑥 = 0, 0, (𝑁↑(𝑃 pCnt 𝑥))) = if((𝑦 + 𝑧) = 0, 0, (𝑁↑(𝑃 pCnt (𝑦 + 𝑧)))))
204		ovex 7390	. . . . . 6 ⊢ (𝑁↑(𝑃 pCnt (𝑦 + 𝑧))) ∈ V
205	42, 204	ifex 4536	. . . . 5 ⊢ if((𝑦 + 𝑧) = 0, 0, (𝑁↑(𝑃 pCnt (𝑦 + 𝑧)))) ∈ V
206	203, 36, 205	fvmpt 6948	. . . 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 7375	. . 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 5137	. 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 20279	1 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (0(,)1)) → 𝐹 ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 Vcvv 3445 ifcif 4486 class class class wbr 5105 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 < clt 11189 ≤ cle 11190 ℤcz 12499 ℚcq 12873 ℝ⁺crp 12915 (,)cioo 13264 ↑cexp 13967 ℙcprime 16547 pCnt cpc 16708 Basecbs 17083 ↾_s cress 17112 +_gcplusg 17133 ._rcmulr 17134 0_gc0g 17321 Ringcrg 19964 DivRingcdr 20185 AbsValcabv 20275 ℂ_fldccnfld 20796

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-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 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 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

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-rmo 3353 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-tp 4591 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-1st 7921 df-2nd 7922 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-ioo 13268 df-ico 13270 df-fz 13425 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-dvds 16137 df-gcd 16375 df-prm 16548 df-pc 16709 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-0g 17323 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-grp 18751 df-minusg 18752 df-subg 18925 df-cmn 19564 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-drng 20187 df-subrg 20220 df-abv 20276 df-cnfld 20797

This theorem is referenced by: padicabvf 26979 padicabvcxp 26980

W3C validator