Theorem dvdsrval 20309

Description: Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
dvdsr.1	⊢ 𝐵 = (Base‘𝑅)
dvdsr.2	⊢ ∥ = (∥r‘𝑅)
dvdsr.3	⊢ · = (.r‘𝑅)

Assertion

Ref	Expression
dvdsrval	⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)}

Distinct variable groups:   𝑥,𝑦, ∥   𝑥,𝑧,𝐵,𝑦   𝑥,𝑅,𝑦,𝑧   𝑥, · ,𝑦,𝑧

Allowed substitution hint:   ∥ (𝑧)

Proof of Theorem dvdsrval

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvdsr.2	. . 3 ⊢ ∥ = (∥r‘𝑅)
2		fveq2 6842	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3		dvdsr.1	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	eqtr4di 2790	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5	4	eleq2d 2823	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥 ∈ 𝐵))
6	4	rexeqdv 3299	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦 ↔ ∃𝑧 ∈ 𝐵 (𝑧(.r‘𝑟)𝑥) = 𝑦))
7	5, 6	anbi12d 633	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧(.r‘𝑟)𝑥) = 𝑦)))
8		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
9		dvdsr.3	. . . . . . . . . . 11 ⊢ · = (.r‘𝑅)
10	8, 9	eqtr4di 2790	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
11	10	oveqd 7385	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑧(.r‘𝑟)𝑥) = (𝑧 · 𝑥))
12	11	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑧(.r‘𝑟)𝑥) = 𝑦 ↔ (𝑧 · 𝑥) = 𝑦))
13	12	rexbidv 3162	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∃𝑧 ∈ 𝐵 (𝑧(.r‘𝑟)𝑥) = 𝑦 ↔ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦))
14	13	anbi2d 631	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧(.r‘𝑟)𝑥) = 𝑦) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)))
15	7, 14	bitrd 279	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)))
16	15	opabbidv 5166	. . . 4 ⊢ (𝑟 = 𝑅 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)})
17		df-dvdsr 20305	. . . 4 ⊢ ∥r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦)})
18	3	fvexi 6856	. . . . 5 ⊢ 𝐵 ∈ V
19		eqcom 2744	. . . . . . . . 9 ⊢ ((𝑧 · 𝑥) = 𝑦 ↔ 𝑦 = (𝑧 · 𝑥))
20	19	rexbii 3085	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧 ∈ 𝐵 𝑦 = (𝑧 · 𝑥))
21	20	abbii 2804	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦} = {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑦 = (𝑧 · 𝑥)}
22	18	abrexex 7916	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑦 = (𝑧 · 𝑥)} ∈ V
23	21, 22	eqeltri 2833	. . . . . 6 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦} ∈ V
24	23	a1i 11	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → {𝑦 ∣ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦} ∈ V)
25	18, 24	opabex3 7921	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} ∈ V
26	16, 17, 25	fvmpt 6949	. . 3 ⊢ (𝑅 ∈ V → (∥r‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)})
27	1, 26	eqtrid 2784	. 2 ⊢ (𝑅 ∈ V → ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)})
28		fvprc 6834	. . . 4 ⊢ (¬ 𝑅 ∈ V → (∥r‘𝑅) = ∅)
29	1, 28	eqtrid 2784	. . 3 ⊢ (¬ 𝑅 ∈ V → ∥ = ∅)
30		opabn0 5509	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} ≠ ∅ ↔ ∃𝑥∃𝑦(𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦))
31		n0i 4294	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → ¬ 𝐵 = ∅)
32		fvprc 6834	. . . . . . . . 9 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅)
33	3, 32	eqtrid 2784	. . . . . . . 8 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
34	31, 33	nsyl2 141	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → 𝑅 ∈ V)
35	34	adantr 480	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦) → 𝑅 ∈ V)
36	35	exlimivv 1934	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦) → 𝑅 ∈ V)
37	30, 36	sylbi 217	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} ≠ ∅ → 𝑅 ∈ V)
38	37	necon1bi 2961	. . 3 ⊢ (¬ 𝑅 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} = ∅)
39	29, 38	eqtr4d 2775	. 2 ⊢ (¬ 𝑅 ∈ V → ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)})
40	27, 39	pm2.61i 182	1 ⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)}

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∃wrex 3062  Vcvv 3442  ∅c0 4287  {copab 5162  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  ._rcmulr 17190  ∥_rcdsr 20302

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-dvdsr 20305

This theorem is referenced by:  dvdsr  20310  dvdsrpropd  20364  dvdsrzring  21428