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Theorem pceulem 16865

Description: Lemma for pceu 16866. (Contributed by Mario Carneiro, 23-Feb-2014.)

**Hypotheses**
Ref	Expression
pcval.1	⊢ 𝑆 = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < )
pcval.2	⊢ 𝑇 = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )
pceu.3	⊢ 𝑈 = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < )
pceu.4	⊢ 𝑉 = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )
pceu.5	⊢ (𝜑 → 𝑃 ∈ ℙ)
pceu.6	⊢ (𝜑 → 𝑁 ≠ 0)
pceu.7	⊢ (𝜑 → (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ))
pceu.8	⊢ (𝜑 → 𝑁 = (𝑥 / 𝑦))
pceu.9	⊢ (𝜑 → (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ))
pceu.10	⊢ (𝜑 → 𝑁 = (𝑠 / 𝑡))

**Assertion**
Ref	Expression
pceulem	⊢ (𝜑 → (𝑆 − 𝑇) = (𝑈 − 𝑉))

Distinct variable groups: 𝑛,𝑠,𝑡,𝑥,𝑦,𝑁 𝑃,𝑛,𝑠,𝑡,𝑥,𝑦 𝑆,𝑠,𝑡 𝑇,𝑠,𝑡 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑡,𝑛,𝑠) 𝑆(𝑥,𝑦,𝑛) 𝑇(𝑥,𝑦,𝑛) 𝑈(𝑥,𝑦,𝑡,𝑛,𝑠) 𝑉(𝑥,𝑦,𝑡,𝑛,𝑠)

Proof of Theorem pceulem Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pceu.7	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ))
2	1	simprd 495	. . . . . . . . . 10 ⊢ (𝜑 → 𝑦 ∈ ℕ)
3	2	nncnd 12256	. . . . . . . . 9 ⊢ (𝜑 → 𝑦 ∈ ℂ)
4		pceu.9	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ))
5	4	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝑠 ∈ ℤ)
6	5	zcnd 12698	. . . . . . . . 9 ⊢ (𝜑 → 𝑠 ∈ ℂ)
7	3, 6	mulcomd 11256	. . . . . . . 8 ⊢ (𝜑 → (𝑦 · 𝑠) = (𝑠 · 𝑦))
8		pceu.10	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 = (𝑠 / 𝑡))
9		pceu.8	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 = (𝑥 / 𝑦))
10	8, 9	eqtr3d 2772	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 / 𝑡) = (𝑥 / 𝑦))
11	4	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑡 ∈ ℕ)
12	11	nncnd 12256	. . . . . . . . . 10 ⊢ (𝜑 → 𝑡 ∈ ℂ)
13	1	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑥 ∈ ℤ)
14	13	zcnd 12698	. . . . . . . . . 10 ⊢ (𝜑 → 𝑥 ∈ ℂ)
15	11	nnne0d 12290	. . . . . . . . . 10 ⊢ (𝜑 → 𝑡 ≠ 0)
16	2	nnne0d 12290	. . . . . . . . . 10 ⊢ (𝜑 → 𝑦 ≠ 0)
17	6, 12, 14, 3, 15, 16	divmuleqd 12063	. . . . . . . . 9 ⊢ (𝜑 → ((𝑠 / 𝑡) = (𝑥 / 𝑦) ↔ (𝑠 · 𝑦) = (𝑥 · 𝑡)))
18	10, 17	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → (𝑠 · 𝑦) = (𝑥 · 𝑡))
19	7, 18	eqtrd 2770	. . . . . . 7 ⊢ (𝜑 → (𝑦 · 𝑠) = (𝑥 · 𝑡))
20	19	breq2d 5131	. . . . . 6 ⊢ (𝜑 → ((𝑃↑𝑧) ∥ (𝑦 · 𝑠) ↔ (𝑃↑𝑧) ∥ (𝑥 · 𝑡)))
21	20	rabbidv 3423	. . . . 5 ⊢ (𝜑 → {𝑧 ∈ ℕ₀ ∣ (𝑃↑𝑧) ∥ (𝑦 · 𝑠)} = {𝑧 ∈ ℕ₀ ∣ (𝑃↑𝑧) ∥ (𝑥 · 𝑡)})
22		oveq2 7413	. . . . . . 7 ⊢ (𝑛 = 𝑧 → (𝑃↑𝑛) = (𝑃↑𝑧))
23	22	breq1d 5129	. . . . . 6 ⊢ (𝑛 = 𝑧 → ((𝑃↑𝑛) ∥ (𝑦 · 𝑠) ↔ (𝑃↑𝑧) ∥ (𝑦 · 𝑠)))
24	23	cbvrabv 3426	. . . . 5 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)} = {𝑧 ∈ ℕ₀ ∣ (𝑃↑𝑧) ∥ (𝑦 · 𝑠)}
25	22	breq1d 5129	. . . . . 6 ⊢ (𝑛 = 𝑧 → ((𝑃↑𝑛) ∥ (𝑥 · 𝑡) ↔ (𝑃↑𝑧) ∥ (𝑥 · 𝑡)))
26	25	cbvrabv 3426	. . . . 5 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)} = {𝑧 ∈ ℕ₀ ∣ (𝑃↑𝑧) ∥ (𝑥 · 𝑡)}
27	21, 24, 26	3eqtr4g 2795	. . . 4 ⊢ (𝜑 → {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)} = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)})
28	27	supeq1d 9458	. . 3 ⊢ (𝜑 → sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)}, ℝ, < ) = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)}, ℝ, < ))
29		pceu.5	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ℙ)
30	2	nnzd 12615	. . . 4 ⊢ (𝜑 → 𝑦 ∈ ℤ)
31		pceu.6	. . . . 5 ⊢ (𝜑 → 𝑁 ≠ 0)
32	12, 15	div0d 12016	. . . . . . . 8 ⊢ (𝜑 → (0 / 𝑡) = 0)
33		oveq1 7412	. . . . . . . . 9 ⊢ (𝑠 = 0 → (𝑠 / 𝑡) = (0 / 𝑡))
34	33	eqeq1d 2737	. . . . . . . 8 ⊢ (𝑠 = 0 → ((𝑠 / 𝑡) = 0 ↔ (0 / 𝑡) = 0))
35	32, 34	syl5ibrcom 247	. . . . . . 7 ⊢ (𝜑 → (𝑠 = 0 → (𝑠 / 𝑡) = 0))
36	8	eqeq1d 2737	. . . . . . 7 ⊢ (𝜑 → (𝑁 = 0 ↔ (𝑠 / 𝑡) = 0))
37	35, 36	sylibrd 259	. . . . . 6 ⊢ (𝜑 → (𝑠 = 0 → 𝑁 = 0))
38	37	necon3d 2953	. . . . 5 ⊢ (𝜑 → (𝑁 ≠ 0 → 𝑠 ≠ 0))
39	31, 38	mpd 15	. . . 4 ⊢ (𝜑 → 𝑠 ≠ 0)
40		pcval.2	. . . . 5 ⊢ 𝑇 = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )
41		pceu.3	. . . . 5 ⊢ 𝑈 = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < )
42		eqid 2735	. . . . 5 ⊢ sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)}, ℝ, < ) = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)}, ℝ, < )
43	40, 41, 42	pcpremul 16863	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝑦 ∈ ℤ ∧ 𝑦 ≠ 0) ∧ (𝑠 ∈ ℤ ∧ 𝑠 ≠ 0)) → (𝑇 + 𝑈) = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)}, ℝ, < ))
44	29, 30, 16, 5, 39, 43	syl122anc 1381	. . 3 ⊢ (𝜑 → (𝑇 + 𝑈) = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)}, ℝ, < ))
45	3, 16	div0d 12016	. . . . . . . 8 ⊢ (𝜑 → (0 / 𝑦) = 0)
46		oveq1 7412	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 / 𝑦) = (0 / 𝑦))
47	46	eqeq1d 2737	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑥 / 𝑦) = 0 ↔ (0 / 𝑦) = 0))
48	45, 47	syl5ibrcom 247	. . . . . . 7 ⊢ (𝜑 → (𝑥 = 0 → (𝑥 / 𝑦) = 0))
49	9	eqeq1d 2737	. . . . . . 7 ⊢ (𝜑 → (𝑁 = 0 ↔ (𝑥 / 𝑦) = 0))
50	48, 49	sylibrd 259	. . . . . 6 ⊢ (𝜑 → (𝑥 = 0 → 𝑁 = 0))
51	50	necon3d 2953	. . . . 5 ⊢ (𝜑 → (𝑁 ≠ 0 → 𝑥 ≠ 0))
52	31, 51	mpd 15	. . . 4 ⊢ (𝜑 → 𝑥 ≠ 0)
53	11	nnzd 12615	. . . 4 ⊢ (𝜑 → 𝑡 ∈ ℤ)
54		pcval.1	. . . . 5 ⊢ 𝑆 = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < )
55		pceu.4	. . . . 5 ⊢ 𝑉 = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )
56		eqid 2735	. . . . 5 ⊢ sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)}, ℝ, < ) = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)}, ℝ, < )
57	54, 55, 56	pcpremul 16863	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0) ∧ (𝑡 ∈ ℤ ∧ 𝑡 ≠ 0)) → (𝑆 + 𝑉) = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)}, ℝ, < ))
58	29, 13, 52, 53, 15, 57	syl122anc 1381	. . 3 ⊢ (𝜑 → (𝑆 + 𝑉) = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)}, ℝ, < ))
59	28, 44, 58	3eqtr4d 2780	. 2 ⊢ (𝜑 → (𝑇 + 𝑈) = (𝑆 + 𝑉))
60		prmuz2 16715	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
61	29, 60	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (ℤ_≥‘2))
62		eqid 2735	. . . . . . 7 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑦}
63	62, 40	pcprecl 16859	. . . . . 6 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑦 ∈ ℤ ∧ 𝑦 ≠ 0)) → (𝑇 ∈ ℕ₀ ∧ (𝑃↑𝑇) ∥ 𝑦))
64	63	simpld 494	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑦 ∈ ℤ ∧ 𝑦 ≠ 0)) → 𝑇 ∈ ℕ₀)
65	61, 30, 16, 64	syl12anc 836	. . . 4 ⊢ (𝜑 → 𝑇 ∈ ℕ₀)
66	65	nn0cnd 12564	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℂ)
67		eqid 2735	. . . . . . 7 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑠} = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑠}
68	67, 41	pcprecl 16859	. . . . . 6 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑠 ∈ ℤ ∧ 𝑠 ≠ 0)) → (𝑈 ∈ ℕ₀ ∧ (𝑃↑𝑈) ∥ 𝑠))
69	68	simpld 494	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑠 ∈ ℤ ∧ 𝑠 ≠ 0)) → 𝑈 ∈ ℕ₀)
70	61, 5, 39, 69	syl12anc 836	. . . 4 ⊢ (𝜑 → 𝑈 ∈ ℕ₀)
71	70	nn0cnd 12564	. . 3 ⊢ (𝜑 → 𝑈 ∈ ℂ)
72		eqid 2735	. . . . . . 7 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑥}
73	72, 54	pcprecl 16859	. . . . . 6 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → (𝑆 ∈ ℕ₀ ∧ (𝑃↑𝑆) ∥ 𝑥))
74	73	simpld 494	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → 𝑆 ∈ ℕ₀)
75	61, 13, 52, 74	syl12anc 836	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℕ₀)
76	75	nn0cnd 12564	. . 3 ⊢ (𝜑 → 𝑆 ∈ ℂ)
77		eqid 2735	. . . . . . 7 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑡} = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑡}
78	77, 55	pcprecl 16859	. . . . . 6 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑡 ∈ ℤ ∧ 𝑡 ≠ 0)) → (𝑉 ∈ ℕ₀ ∧ (𝑃↑𝑉) ∥ 𝑡))
79	78	simpld 494	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑡 ∈ ℤ ∧ 𝑡 ≠ 0)) → 𝑉 ∈ ℕ₀)
80	61, 53, 15, 79	syl12anc 836	. . . 4 ⊢ (𝜑 → 𝑉 ∈ ℕ₀)
81	80	nn0cnd 12564	. . 3 ⊢ (𝜑 → 𝑉 ∈ ℂ)
82	66, 71, 76, 81	addsubeq4d 11645	. 2 ⊢ (𝜑 → ((𝑇 + 𝑈) = (𝑆 + 𝑉) ↔ (𝑆 − 𝑇) = (𝑈 − 𝑉)))
83	59, 82	mpbid 232	1 ⊢ (𝜑 → (𝑆 − 𝑇) = (𝑈 − 𝑉))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 {crab 3415 class class class wbr 5119 ‘cfv 6531 (class class class)co 7405 supcsup 9452 ℝcr 11128 0cc0 11129 + caddc 11132 · cmul 11134 < clt 11269 − cmin 11466 / cdiv 11894 ℕcn 12240 2c2 12295 ℕ₀cn0 12501 ℤcz 12588 ℤ_≥cuz 12852 ↑cexp 14079 ∥ cdvds 16272 ℙcprime 16690

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-sup 9454 df-inf 9455 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-n0 12502 df-z 12589 df-uz 12853 df-rp 13009 df-fl 13809 df-mod 13887 df-seq 14020 df-exp 14080 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-dvds 16273 df-gcd 16514 df-prm 16691

This theorem is referenced by: pceu 16866

W3C validator