Theorem dvdsrval 20624

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 6911	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3		dvdsr.1	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	eqtr4di 2816	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5	4	eleq2d 2850	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥 ∈ 𝐵))
6	4	rexeqdv 3324	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦 ↔ ∃𝑧 ∈ 𝐵 (𝑧(.r‘𝑟)𝑥) = 𝑦))
7	5, 6	anbi12d 644	. . . . . 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 6911	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
9		dvdsr.3	. . . . . . . . . . 11 ⊢ · = (.r‘𝑅)
10	8, 9	eqtr4di 2816	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
11	10	oveqd 7463	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑧(.r‘𝑟)𝑥) = (𝑧 · 𝑥))
12	11	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑧(.r‘𝑟)𝑥) = 𝑦 ↔ (𝑧 · 𝑥) = 𝑦))
13	12	rexbidv 3190	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∃𝑧 ∈ 𝐵 (𝑧(.r‘𝑟)𝑥) = 𝑦 ↔ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦))
14	13	anbi2d 642	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧(.r‘𝑟)𝑥) = 𝑦) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)))
15	7, 14	bitrd 282	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)))
16	15	opabbidv 5175	. . . 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 20620	. . . 4 ⊢ ∥r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦)})
18	3	fvexi 6925	. . . . 5 ⊢ 𝐵 ∈ V
19		eqcom 2770	. . . . . . . . 9 ⊢ ((𝑧 · 𝑥) = 𝑦 ↔ 𝑦 = (𝑧 · 𝑥))
20	19	rexbii 3113	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧 ∈ 𝐵 𝑦 = (𝑧 · 𝑥))
21	20	abbii 2830	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦} = {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑦 = (𝑧 · 𝑥)}
22	18	abrexex 8001	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑦 = (𝑧 · 𝑥)} ∈ V
23	21, 22	eqeltri 2860	. . . . . 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 8006	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} ∈ V
26	16, 17, 25	fvmpt 7019	. . 3 ⊢ (𝑅 ∈ V → (∥r‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)})
27	1, 26	eqtrid 2810	. 2 ⊢ (𝑅 ∈ V → ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)})
28		fvprc 6903	. . . 4 ⊢ (¬ 𝑅 ∈ V → (∥r‘𝑅) = ∅)
29	1, 28	eqtrid 2810	. . 3 ⊢ (¬ 𝑅 ∈ V → ∥ = ∅)
30		opabn0 5533	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} ≠ ∅ ↔ ∃𝑥∃𝑦(𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦))
31		n0i 4289	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → ¬ 𝐵 = ∅)
32		fvprc 6903	. . . . . . . . 9 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅)
33	3, 32	eqtrid 2810	. . . . . . . 8 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
34	31, 33	nsyl2 142	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → 𝑅 ∈ V)
35	34	adantr 486	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦) → 𝑅 ∈ V)
36	35	exlimivv 1967	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦) → 𝑅 ∈ V)
37	30, 36	sylbi 220	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} ≠ ∅ → 𝑅 ∈ V)
38	37	necon1bi 2987	. . 3 ⊢ (¬ 𝑅 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} = ∅)
39	29, 38	eqtr4d 2801	. 2 ⊢ (¬ 𝑅 ∈ V → ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)})
40	27, 39	pm2.61i 184	1 ⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)}

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∃wrex 3090  Vcvv 3454  ∅c0 4281  {copab 5171  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  ._rcmulr 17461  ∥_rcdsr 20617

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-iota 6515  df-fun 6563  df-fv 6569  df-ov 7449  df-dvdsr 20620

This theorem is used by:  dvdsr  20625  dvdsrpropd  20679  dvdsrzring  21800