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Theorem pceulem 15388

Description: Lemma for pceu 15389. (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 478	. . . . . . . . . 10 ⊢ (𝜑 → 𝑦 ∈ ℕ)
3	2	nncnd 10913	. . . . . . . . 9 ⊢ (𝜑 → 𝑦 ∈ ℂ)
4		pceu.9	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ))
5	4	simpld 474	. . . . . . . . . 10 ⊢ (𝜑 → 𝑠 ∈ ℤ)
6	5	zcnd 11359	. . . . . . . . 9 ⊢ (𝜑 → 𝑠 ∈ ℂ)
7	3, 6	mulcomd 9940	. . . . . . . 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 2646	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 / 𝑡) = (𝑥 / 𝑦))
11	4	simprd 478	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑡 ∈ ℕ)
12	11	nncnd 10913	. . . . . . . . . 10 ⊢ (𝜑 → 𝑡 ∈ ℂ)
13	1	simpld 474	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑥 ∈ ℤ)
14	13	zcnd 11359	. . . . . . . . . 10 ⊢ (𝜑 → 𝑥 ∈ ℂ)
15	11	nnne0d 10942	. . . . . . . . . 10 ⊢ (𝜑 → 𝑡 ≠ 0)
16	2	nnne0d 10942	. . . . . . . . . 10 ⊢ (𝜑 → 𝑦 ≠ 0)
17	6, 12, 14, 3, 15, 16	divmuleqd 10726	. . . . . . . . 9 ⊢ (𝜑 → ((𝑠 / 𝑡) = (𝑥 / 𝑦) ↔ (𝑠 · 𝑦) = (𝑥 · 𝑡)))
18	10, 17	mpbid 221	. . . . . . . 8 ⊢ (𝜑 → (𝑠 · 𝑦) = (𝑥 · 𝑡))
19	7, 18	eqtrd 2644	. . . . . . 7 ⊢ (𝜑 → (𝑦 · 𝑠) = (𝑥 · 𝑡))
20	19	breq2d 4595	. . . . . 6 ⊢ (𝜑 → ((𝑃↑𝑧) ∥ (𝑦 · 𝑠) ↔ (𝑃↑𝑧) ∥ (𝑥 · 𝑡)))
21	20	rabbidv 3164	. . . . 5 ⊢ (𝜑 → {𝑧 ∈ ℕ₀ ∣ (𝑃↑𝑧) ∥ (𝑦 · 𝑠)} = {𝑧 ∈ ℕ₀ ∣ (𝑃↑𝑧) ∥ (𝑥 · 𝑡)})
22		oveq2 6557	. . . . . . 7 ⊢ (𝑛 = 𝑧 → (𝑃↑𝑛) = (𝑃↑𝑧))
23	22	breq1d 4593	. . . . . 6 ⊢ (𝑛 = 𝑧 → ((𝑃↑𝑛) ∥ (𝑦 · 𝑠) ↔ (𝑃↑𝑧) ∥ (𝑦 · 𝑠)))
24	23	cbvrabv 3172	. . . . 5 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)} = {𝑧 ∈ ℕ₀ ∣ (𝑃↑𝑧) ∥ (𝑦 · 𝑠)}
25	22	breq1d 4593	. . . . . 6 ⊢ (𝑛 = 𝑧 → ((𝑃↑𝑛) ∥ (𝑥 · 𝑡) ↔ (𝑃↑𝑧) ∥ (𝑥 · 𝑡)))
26	25	cbvrabv 3172	. . . . 5 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)} = {𝑧 ∈ ℕ₀ ∣ (𝑃↑𝑧) ∥ (𝑥 · 𝑡)}
27	21, 24, 26	3eqtr4g 2669	. . . 4 ⊢ (𝜑 → {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)} = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)})
28	27	supeq1d 8235	. . 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 11357	. . . 4 ⊢ (𝜑 → 𝑦 ∈ ℤ)
31		pceu.6	. . . . 5 ⊢ (𝜑 → 𝑁 ≠ 0)
32	12, 15	div0d 10679	. . . . . . . 8 ⊢ (𝜑 → (0 / 𝑡) = 0)
33		oveq1 6556	. . . . . . . . 9 ⊢ (𝑠 = 0 → (𝑠 / 𝑡) = (0 / 𝑡))
34	33	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑠 = 0 → ((𝑠 / 𝑡) = 0 ↔ (0 / 𝑡) = 0))
35	32, 34	syl5ibrcom 236	. . . . . . 7 ⊢ (𝜑 → (𝑠 = 0 → (𝑠 / 𝑡) = 0))
36	8	eqeq1d 2612	. . . . . . 7 ⊢ (𝜑 → (𝑁 = 0 ↔ (𝑠 / 𝑡) = 0))
37	35, 36	sylibrd 248	. . . . . 6 ⊢ (𝜑 → (𝑠 = 0 → 𝑁 = 0))
38	37	necon3d 2803	. . . . 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 2610	. . . . 5 ⊢ sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)}, ℝ, < ) = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)}, ℝ, < )
43	40, 41, 42	pcpremul 15386	. . . 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 1327	. . 3 ⊢ (𝜑 → (𝑇 + 𝑈) = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑦 · 𝑠)}, ℝ, < ))
45	3, 16	div0d 10679	. . . . . . . 8 ⊢ (𝜑 → (0 / 𝑦) = 0)
46		oveq1 6556	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 / 𝑦) = (0 / 𝑦))
47	46	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑥 / 𝑦) = 0 ↔ (0 / 𝑦) = 0))
48	45, 47	syl5ibrcom 236	. . . . . . 7 ⊢ (𝜑 → (𝑥 = 0 → (𝑥 / 𝑦) = 0))
49	9	eqeq1d 2612	. . . . . . 7 ⊢ (𝜑 → (𝑁 = 0 ↔ (𝑥 / 𝑦) = 0))
50	48, 49	sylibrd 248	. . . . . 6 ⊢ (𝜑 → (𝑥 = 0 → 𝑁 = 0))
51	50	necon3d 2803	. . . . 5 ⊢ (𝜑 → (𝑁 ≠ 0 → 𝑥 ≠ 0))
52	31, 51	mpd 15	. . . 4 ⊢ (𝜑 → 𝑥 ≠ 0)
53	11	nnzd 11357	. . . 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 2610	. . . . 5 ⊢ sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)}, ℝ, < ) = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)}, ℝ, < )
57	54, 55, 56	pcpremul 15386	. . . 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 1327	. . 3 ⊢ (𝜑 → (𝑆 + 𝑉) = sup({𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ (𝑥 · 𝑡)}, ℝ, < ))
59	28, 44, 58	3eqtr4d 2654	. 2 ⊢ (𝜑 → (𝑇 + 𝑈) = (𝑆 + 𝑉))
60		prmuz2 15246	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
61	29, 60	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (ℤ_≥‘2))
62		eqid 2610	. . . . . . 7 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑦}
63	62, 40	pcprecl 15382	. . . . . 6 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑦 ∈ ℤ ∧ 𝑦 ≠ 0)) → (𝑇 ∈ ℕ₀ ∧ (𝑃↑𝑇) ∥ 𝑦))
64	63	simpld 474	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑦 ∈ ℤ ∧ 𝑦 ≠ 0)) → 𝑇 ∈ ℕ₀)
65	61, 30, 16, 64	syl12anc 1316	. . . 4 ⊢ (𝜑 → 𝑇 ∈ ℕ₀)
66	65	nn0cnd 11230	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℂ)
67		eqid 2610	. . . . . . 7 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑠} = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑠}
68	67, 41	pcprecl 15382	. . . . . 6 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑠 ∈ ℤ ∧ 𝑠 ≠ 0)) → (𝑈 ∈ ℕ₀ ∧ (𝑃↑𝑈) ∥ 𝑠))
69	68	simpld 474	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑠 ∈ ℤ ∧ 𝑠 ≠ 0)) → 𝑈 ∈ ℕ₀)
70	61, 5, 39, 69	syl12anc 1316	. . . 4 ⊢ (𝜑 → 𝑈 ∈ ℕ₀)
71	70	nn0cnd 11230	. . 3 ⊢ (𝜑 → 𝑈 ∈ ℂ)
72		eqid 2610	. . . . . . 7 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑥}
73	72, 54	pcprecl 15382	. . . . . 6 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → (𝑆 ∈ ℕ₀ ∧ (𝑃↑𝑆) ∥ 𝑥))
74	73	simpld 474	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → 𝑆 ∈ ℕ₀)
75	61, 13, 52, 74	syl12anc 1316	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℕ₀)
76	75	nn0cnd 11230	. . 3 ⊢ (𝜑 → 𝑆 ∈ ℂ)
77		eqid 2610	. . . . . . 7 ⊢ {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑡} = {𝑛 ∈ ℕ₀ ∣ (𝑃↑𝑛) ∥ 𝑡}
78	77, 55	pcprecl 15382	. . . . . 6 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑡 ∈ ℤ ∧ 𝑡 ≠ 0)) → (𝑉 ∈ ℕ₀ ∧ (𝑃↑𝑉) ∥ 𝑡))
79	78	simpld 474	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ (𝑡 ∈ ℤ ∧ 𝑡 ≠ 0)) → 𝑉 ∈ ℕ₀)
80	61, 53, 15, 79	syl12anc 1316	. . . 4 ⊢ (𝜑 → 𝑉 ∈ ℕ₀)
81	80	nn0cnd 11230	. . 3 ⊢ (𝜑 → 𝑉 ∈ ℂ)
82	66, 71, 76, 81	addsubeq4d 10322	. 2 ⊢ (𝜑 → ((𝑇 + 𝑈) = (𝑆 + 𝑉) ↔ (𝑆 − 𝑇) = (𝑈 − 𝑉)))
83	59, 82	mpbid 221	1 ⊢ (𝜑 → (𝑆 − 𝑇) = (𝑈 − 𝑉))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 supcsup 8229 ℝcr 9814 0cc0 9815 + caddc 9818 · cmul 9820 < clt 9953 − cmin 10145 / cdiv 10563 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ↑cexp 12722 ∥ cdvds 14821 ℙcprime 15223

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-gcd 15055 df-prm 15224

This theorem is referenced by: pceu 15389

W3C validator