Theorem dvdsrval 18645

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 6191	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3		dvdsr.1	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5	4	eleq2d 2687	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥 ∈ 𝐵))
6	4	rexeqdv 3145	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦 ↔ ∃𝑧 ∈ 𝐵 (𝑧(.r‘𝑟)𝑥) = 𝑦))
7	5, 6	anbi12d 747	. . . . . 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 6191	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
9		dvdsr.3	. . . . . . . . . . 11 ⊢ · = (.r‘𝑅)
10	8, 9	syl6eqr 2674	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
11	10	oveqd 6667	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑧(.r‘𝑟)𝑥) = (𝑧 · 𝑥))
12	11	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑧(.r‘𝑟)𝑥) = 𝑦 ↔ (𝑧 · 𝑥) = 𝑦))
13	12	rexbidv 3052	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∃𝑧 ∈ 𝐵 (𝑧(.r‘𝑟)𝑥) = 𝑦 ↔ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦))
14	13	anbi2d 740	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧(.r‘𝑟)𝑥) = 𝑦) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)))
15	7, 14	bitrd 268	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)))
16	15	opabbidv 4716	. . . 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 18641	. . . 4 ⊢ ∥r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦)})
18		fvex 6201	. . . . . 6 ⊢ (Base‘𝑅) ∈ V
19	3, 18	eqeltri 2697	. . . . 5 ⊢ 𝐵 ∈ V
20		eqcom 2629	. . . . . . . . 9 ⊢ ((𝑧 · 𝑥) = 𝑦 ↔ 𝑦 = (𝑧 · 𝑥))
21	20	rexbii 3041	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧 ∈ 𝐵 𝑦 = (𝑧 · 𝑥))
22	21	abbii 2739	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦} = {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑦 = (𝑧 · 𝑥)}
23	19	abrexex 7141	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑦 = (𝑧 · 𝑥)} ∈ V
24	22, 23	eqeltri 2697	. . . . . 6 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦} ∈ V
25	24	a1i 11	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → {𝑦 ∣ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦} ∈ V)
26	19, 25	opabex3 7146	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} ∈ V
27	16, 17, 26	fvmpt 6282	. . 3 ⊢ (𝑅 ∈ V → (∥r‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)})
28	1, 27	syl5eq 2668	. 2 ⊢ (𝑅 ∈ V → ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)})
29		fvprc 6185	. . . 4 ⊢ (¬ 𝑅 ∈ V → (∥r‘𝑅) = ∅)
30	1, 29	syl5eq 2668	. . 3 ⊢ (¬ 𝑅 ∈ V → ∥ = ∅)
31		opabn0 5006	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} ≠ ∅ ↔ ∃𝑥∃𝑦(𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦))
32		n0i 3920	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → ¬ 𝐵 = ∅)
33		fvprc 6185	. . . . . . . . 9 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅)
34	3, 33	syl5eq 2668	. . . . . . . 8 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
35	32, 34	nsyl2 142	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → 𝑅 ∈ V)
36	35	adantr 481	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦) → 𝑅 ∈ V)
37	36	exlimivv 1860	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦) → 𝑅 ∈ V)
38	31, 37	sylbi 207	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} ≠ ∅ → 𝑅 ∈ V)
39	38	necon1bi 2822	. . 3 ⊢ (¬ 𝑅 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} = ∅)
40	30, 39	eqtr4d 2659	. 2 ⊢ (¬ 𝑅 ∈ V → ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)})
41	28, 40	pm2.61i 176	1 ⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)}

Syntax hints:  ¬ wn 3   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∃wrex 2913  Vcvv 3200  ∅c0 3915  {copab 4712  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  ._rcmulr 15942  ∥_rcdsr 18638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-dvdsr 18641

This theorem is referenced by:  dvdsr  18646  dvdsrpropd  18696  dvdsrzring  19831