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Theorem pceulem 16892

Description: Lemma for pceu 16893. (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 12309	. . . . . . . . 9 ⊢ (𝜑 → 𝑦 ∈ ℂ)
4		pceu.9	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ))
5	4	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝑠 ∈ ℤ)
6	5	zcnd 12748	. . . . . . . . 9 ⊢ (𝜑 → 𝑠 ∈ ℂ)
7	3, 6	mulcomd 11311	. . . . . . . 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 2782	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 / 𝑡) = (𝑥 / 𝑦))
11	4	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑡 ∈ ℕ)
12	11	nncnd 12309	. . . . . . . . . 10 ⊢ (𝜑 → 𝑡 ∈ ℂ)
13	1	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑥 ∈ ℤ)
14	13	zcnd 12748	. . . . . . . . . 10 ⊢ (𝜑 → 𝑥 ∈ ℂ)
15	11	nnne0d 12343	. . . . . . . . . 10 ⊢ (𝜑 → 𝑡 ≠ 0)
16	2	nnne0d 12343	. . . . . . . . . 10 ⊢ (𝜑 → 𝑦 ≠ 0)
17	6, 12, 14, 3, 15, 16	divmuleqd 12116	. . . . . . . . 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 2780	. . . . . . 7 ⊢ (𝜑 → (𝑦 · 𝑠) = (𝑥 · 𝑡))
20	19	breq2d 5178	. . . . . 6 ⊢ (𝜑 → ((𝑃↑𝑧) ∥ (𝑦 · 𝑠) ↔ (𝑃↑𝑧) ∥ (𝑥 · 𝑡)))
21	20	rabbidv 3451	. . . . 5 ⊢ (𝜑 → {𝑧 ∈ ℕ₀ ∣ (𝑃↑𝑧) ∥ (𝑦 · 𝑠)} = {𝑧 ∈ ℕ₀ ∣ (𝑃↑𝑧) ∥ (𝑥 · 𝑡)})
22		oveq2 7456	. . . . . . 7 ⊢ (𝑛 = 𝑧 → (𝑃↑𝑛) = (𝑃↑𝑧))
23	22	breq1d 5176	. . . . . 6 ⊢ (𝑛 = 𝑧 → ((𝑃↑𝑛) ∥ (𝑦 · 𝑠) ↔ (𝑃↑𝑧) ∥ (𝑦 · 𝑠)))
24	23	cbvrabv 3454	. . . . 5 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)} = {𝑧 ∈ ℕ₀ ∣ (𝑃↑𝑧) ∥ (𝑦 · 𝑠)}
25	22	breq1d 5176	. . . . . 6 ⊢ (𝑛 = 𝑧 → ((𝑃↑𝑛) ∥ (𝑥 · 𝑡) ↔ (𝑃↑𝑧) ∥ (𝑥 · 𝑡)))
26	25	cbvrabv 3454	. . . . 5 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)} = {𝑧 ∈ ℕ₀ ∣ (𝑃↑𝑧) ∥ (𝑥 · 𝑡)}
27	21, 24, 26	3eqtr4g 2805	. . . 4 ⊢ (𝜑 → {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)} = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)})
28	27	supeq1d 9515	. . 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 12666	. . . 4 ⊢ (𝜑 → 𝑦 ∈ ℤ)
31		pceu.6	. . . . 5 ⊢ (𝜑 → 𝑁 ≠ 0)
32	12, 15	div0d 12069	. . . . . . . 8 ⊢ (𝜑 → (0 / 𝑡) = 0)
33		oveq1 7455	. . . . . . . . 9 ⊢ (𝑠 = 0 → (𝑠 / 𝑡) = (0 / 𝑡))
34	33	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑠 = 0 → ((𝑠 / 𝑡) = 0 ↔ (0 / 𝑡) = 0))
35	32, 34	syl5ibrcom 247	. . . . . . 7 ⊢ (𝜑 → (𝑠 = 0 → (𝑠 / 𝑡) = 0))
36	8	eqeq1d 2742	. . . . . . 7 ⊢ (𝜑 → (𝑁 = 0 ↔ (𝑠 / 𝑡) = 0))
37	35, 36	sylibrd 259	. . . . . 6 ⊢ (𝜑 → (𝑠 = 0 → 𝑁 = 0))
38	37	necon3d 2967	. . . . 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 2740	. . . . 5 ⊢ sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)}, ℝ, < ) = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)}, ℝ, < )
43	40, 41, 42	pcpremul 16890	. . . 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 1379	. . 3 ⊢ (𝜑 → (𝑇 + 𝑈) = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)}, ℝ, < ))
45	3, 16	div0d 12069	. . . . . . . 8 ⊢ (𝜑 → (0 / 𝑦) = 0)
46		oveq1 7455	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 / 𝑦) = (0 / 𝑦))
47	46	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑥 / 𝑦) = 0 ↔ (0 / 𝑦) = 0))
48	45, 47	syl5ibrcom 247	. . . . . . 7 ⊢ (𝜑 → (𝑥 = 0 → (𝑥 / 𝑦) = 0))
49	9	eqeq1d 2742	. . . . . . 7 ⊢ (𝜑 → (𝑁 = 0 ↔ (𝑥 / 𝑦) = 0))
50	48, 49	sylibrd 259	. . . . . 6 ⊢ (𝜑 → (𝑥 = 0 → 𝑁 = 0))
51	50	necon3d 2967	. . . . 5 ⊢ (𝜑 → (𝑁 ≠ 0 → 𝑥 ≠ 0))
52	31, 51	mpd 15	. . . 4 ⊢ (𝜑 → 𝑥 ≠ 0)
53	11	nnzd 12666	. . . 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 2740	. . . . 5 ⊢ sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)}, ℝ, < ) = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)}, ℝ, < )
57	54, 55, 56	pcpremul 16890	. . . 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 1379	. . 3 ⊢ (𝜑 → (𝑆 + 𝑉) = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)}, ℝ, < ))
59	28, 44, 58	3eqtr4d 2790	. 2 ⊢ (𝜑 → (𝑇 + 𝑈) = (𝑆 + 𝑉))
60		prmuz2 16743	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
61	29, 60	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (ℤ_≥‘2))
62		eqid 2740	. . . . . . 7 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑦}
63	62, 40	pcprecl 16886	. . . . . 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 12615	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℂ)
67		eqid 2740	. . . . . . 7 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑠} = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑠}
68	67, 41	pcprecl 16886	. . . . . 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 12615	. . 3 ⊢ (𝜑 → 𝑈 ∈ ℂ)
72		eqid 2740	. . . . . . 7 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑥}
73	72, 54	pcprecl 16886	. . . . . 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 12615	. . 3 ⊢ (𝜑 → 𝑆 ∈ ℂ)
77		eqid 2740	. . . . . . 7 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑡} = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑡}
78	77, 55	pcprecl 16886	. . . . . 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 12615	. . 3 ⊢ (𝜑 → 𝑉 ∈ ℂ)
82	66, 71, 76, 81	addsubeq4d 11698	. 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 1537 ∈ wcel 2108 ≠ wne 2946 {crab 3443 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 supcsup 9509 ℝcr 11183 0cc0 11184 + caddc 11187 · cmul 11189 < clt 11324 − cmin 11520 / cdiv 11947 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℤcz 12639 ℤ_≥cuz 12903 ↑cexp 14112 ∥ cdvds 16302 ℙcprime 16718

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-dvds 16303 df-gcd 16541 df-prm 16719

This theorem is referenced by: pceu 16893

W3C validator