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Theorem pceulem 16901

Description: Lemma for pceu 16902. (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 500	. . . . . . . . . 10 ⊢ (𝜑 → 𝑦 ∈ ℕ)
3	2	nncnd 12245	. . . . . . . . 9 ⊢ (𝜑 → 𝑦 ∈ ℂ)
4		pceu.9	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ))
5	4	simpld 499	. . . . . . . . . 10 ⊢ (𝜑 → 𝑠 ∈ ℤ)
6	5	zcnd 12697	. . . . . . . . 9 ⊢ (𝜑 → 𝑠 ∈ ℂ)
7	3, 6	mulcomd 11226	. . . . . . . 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 2806	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 / 𝑡) = (𝑥 / 𝑦))
11	4	simprd 500	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑡 ∈ ℕ)
12	11	nncnd 12245	. . . . . . . . . 10 ⊢ (𝜑 → 𝑡 ∈ ℂ)
13	1	simpld 499	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑥 ∈ ℤ)
14	13	zcnd 12697	. . . . . . . . . 10 ⊢ (𝜑 → 𝑥 ∈ ℂ)
15	11	nnne0d 12282	. . . . . . . . . 10 ⊢ (𝜑 → 𝑡 ≠ 0)
16	2	nnne0d 12282	. . . . . . . . . 10 ⊢ (𝜑 → 𝑦 ≠ 0)
17	6, 12, 14, 3, 15, 16	divmuleqd 12033	. . . . . . . . 9 ⊢ (𝜑 → ((𝑠 / 𝑡) = (𝑥 / 𝑦) ↔ (𝑠 · 𝑦) = (𝑥 · 𝑡)))
18	10, 17	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → (𝑠 · 𝑦) = (𝑥 · 𝑡))
19	7, 18	eqtrd 2804	. . . . . . 7 ⊢ (𝜑 → (𝑦 · 𝑠) = (𝑥 · 𝑡))
20	19	breq2d 5122	. . . . . 6 ⊢ (𝜑 → ((𝑃↑𝑧) ∥ (𝑦 · 𝑠) ↔ (𝑃↑𝑧) ∥ (𝑥 · 𝑡)))
21	20	rabbidv 3430	. . . . 5 ⊢ (𝜑 → {𝑧 ∈ ℕ₀ ∣ (𝑃↑𝑧) ∥ (𝑦 · 𝑠)} = {𝑧 ∈ ℕ₀ ∣ (𝑃↑𝑧) ∥ (𝑥 · 𝑡)})
22		oveq2 7416	. . . . . . 7 ⊢ (𝑛 = 𝑧 → (𝑃↑𝑛) = (𝑃↑𝑧))
23	22	breq1d 5120	. . . . . 6 ⊢ (𝑛 = 𝑧 → ((𝑃↑𝑛) ∥ (𝑦 · 𝑠) ↔ (𝑃↑𝑧) ∥ (𝑦 · 𝑠)))
24	23	cbvrabv 3433	. . . . 5 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)} = {𝑧 ∈ ℕ₀ ∣ (𝑃↑𝑧) ∥ (𝑦 · 𝑠)}
25	22	breq1d 5120	. . . . . 6 ⊢ (𝑛 = 𝑧 → ((𝑃↑𝑛) ∥ (𝑥 · 𝑡) ↔ (𝑃↑𝑧) ∥ (𝑥 · 𝑡)))
26	25	cbvrabv 3433	. . . . 5 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)} = {𝑧 ∈ ℕ₀ ∣ (𝑃↑𝑧) ∥ (𝑥 · 𝑡)}
27	21, 24, 26	3eqtr4g 2829	. . . 4 ⊢ (𝜑 → {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)} = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)})
28	27	supeq1d 9402	. . 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 12613	. . . 4 ⊢ (𝜑 → 𝑦 ∈ ℤ)
31		pceu.6	. . . . 5 ⊢ (𝜑 → 𝑁 ≠ 0)
32	12, 15	div0d 11986	. . . . . . . 8 ⊢ (𝜑 → (0 / 𝑡) = 0)
33		oveq1 7415	. . . . . . . . 9 ⊢ (𝑠 = 0 → (𝑠 / 𝑡) = (0 / 𝑡))
34	33	eqeq1d 2771	. . . . . . . 8 ⊢ (𝑠 = 0 → ((𝑠 / 𝑡) = 0 ↔ (0 / 𝑡) = 0))
35	32, 34	syl5ibrcom 250	. . . . . . 7 ⊢ (𝜑 → (𝑠 = 0 → (𝑠 / 𝑡) = 0))
36	8	eqeq1d 2771	. . . . . . 7 ⊢ (𝜑 → (𝑁 = 0 ↔ (𝑠 / 𝑡) = 0))
37	35, 36	sylibrd 262	. . . . . 6 ⊢ (𝜑 → (𝑠 = 0 → 𝑁 = 0))
38	37	necon3d 2985	. . . . 5 ⊢ (𝜑 → (𝑁 ≠ 0 → 𝑠 ≠ 0))
39	31, 38	mpd 16	. . . 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 2769	. . . . 5 ⊢ sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)}, ℝ, < ) = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)}, ℝ, < )
43	40, 41, 42	pcpremul 16899	. . . 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 1404	. . 3 ⊢ (𝜑 → (𝑇 + 𝑈) = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)}, ℝ, < ))
45	3, 16	div0d 11986	. . . . . . . 8 ⊢ (𝜑 → (0 / 𝑦) = 0)
46		oveq1 7415	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 / 𝑦) = (0 / 𝑦))
47	46	eqeq1d 2771	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑥 / 𝑦) = 0 ↔ (0 / 𝑦) = 0))
48	45, 47	syl5ibrcom 250	. . . . . . 7 ⊢ (𝜑 → (𝑥 = 0 → (𝑥 / 𝑦) = 0))
49	9	eqeq1d 2771	. . . . . . 7 ⊢ (𝜑 → (𝑁 = 0 ↔ (𝑥 / 𝑦) = 0))
50	48, 49	sylibrd 262	. . . . . 6 ⊢ (𝜑 → (𝑥 = 0 → 𝑁 = 0))
51	50	necon3d 2985	. . . . 5 ⊢ (𝜑 → (𝑁 ≠ 0 → 𝑥 ≠ 0))
52	31, 51	mpd 16	. . . 4 ⊢ (𝜑 → 𝑥 ≠ 0)
53	11	nnzd 12613	. . . 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 2769	. . . . 5 ⊢ sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)}, ℝ, < ) = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)}, ℝ, < )
57	54, 55, 56	pcpremul 16899	. . . 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 1404	. . 3 ⊢ (𝜑 → (𝑆 + 𝑉) = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)}, ℝ, < ))
59	28, 44, 58	3eqtr4d 2814	. 2 ⊢ (𝜑 → (𝑇 + 𝑈) = (𝑆 + 𝑉))
60		prmuz2 16750	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
61	29, 60	syl 18	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (ℤ_≥‘2))
62		eqid 2769	. . . . . . 7 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑦}
63	62, 40	pcprecl 16895	. . . . . 6 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑦 ∈ ℤ ∧ 𝑦 ≠ 0)) → (𝑇 ∈ ℕ₀ ∧ (𝑃↑𝑇) ∥ 𝑦))
64	63	simpld 499	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑦 ∈ ℤ ∧ 𝑦 ≠ 0)) → 𝑇 ∈ ℕ₀)
65	61, 30, 16, 64	syl12anc 849	. . . 4 ⊢ (𝜑 → 𝑇 ∈ ℕ₀)
66	65	nn0cnd 12563	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℂ)
67		eqid 2769	. . . . . . 7 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑠} = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑠}
68	67, 41	pcprecl 16895	. . . . . 6 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑠 ∈ ℤ ∧ 𝑠 ≠ 0)) → (𝑈 ∈ ℕ₀ ∧ (𝑃↑𝑈) ∥ 𝑠))
69	68	simpld 499	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑠 ∈ ℤ ∧ 𝑠 ≠ 0)) → 𝑈 ∈ ℕ₀)
70	61, 5, 39, 69	syl12anc 849	. . . 4 ⊢ (𝜑 → 𝑈 ∈ ℕ₀)
71	70	nn0cnd 12563	. . 3 ⊢ (𝜑 → 𝑈 ∈ ℂ)
72		eqid 2769	. . . . . . 7 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑥}
73	72, 54	pcprecl 16895	. . . . . 6 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → (𝑆 ∈ ℕ₀ ∧ (𝑃↑𝑆) ∥ 𝑥))
74	73	simpld 499	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → 𝑆 ∈ ℕ₀)
75	61, 13, 52, 74	syl12anc 849	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℕ₀)
76	75	nn0cnd 12563	. . 3 ⊢ (𝜑 → 𝑆 ∈ ℂ)
77		eqid 2769	. . . . . . 7 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑡} = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑡}
78	77, 55	pcprecl 16895	. . . . . 6 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑡 ∈ ℤ ∧ 𝑡 ≠ 0)) → (𝑉 ∈ ℕ₀ ∧ (𝑃↑𝑉) ∥ 𝑡))
79	78	simpld 499	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑡 ∈ ℤ ∧ 𝑡 ≠ 0)) → 𝑉 ∈ ℕ₀)
80	61, 53, 15, 79	syl12anc 849	. . . 4 ⊢ (𝜑 → 𝑉 ∈ ℕ₀)
81	80	nn0cnd 12563	. . 3 ⊢ (𝜑 → 𝑉 ∈ ℂ)
82	66, 71, 76, 81	addsubeq4d 11616	. 2 ⊢ (𝜑 → ((𝑇 + 𝑈) = (𝑆 + 𝑉) ↔ (𝑆 − 𝑇) = (𝑈 − 𝑉)))
83	59, 82	mpbid 235	1 ⊢ (𝜑 → (𝑆 − 𝑇) = (𝑈 − 𝑉))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 {crab 3423 class class class wbr 5110 ‘cfv 6533 (class class class)co 7408 supcsup 9396 ℝcr 11095 0cc0 11096 + caddc 11099 · cmul 11101 < clt 11239 − cmin 11437 / cdiv 11867 ℕcn 12229 2c2 12291 ℕ₀cn0 12500 ℤcz 12587 ℤ_≥cuz 12858 ↑cexp 14093 ∥ cdvds 16306 ℙcprime 16725

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173 ax-pre-sup 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7859 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-inf 9399 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-div 11868 df-nn 12230 df-2 12299 df-3 12300 df-n0 12501 df-z 12588 df-uz 12859 df-rp 13013 df-fl 13821 df-mod 13899 df-seq 14034 df-exp 14094 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-dvds 16307 df-gcd 16549 df-prm 16726

This theorem is referenced by: pceu 16902

W3C validator