Theorem dvdsrval 20389

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 6863	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3		dvdsr.1	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	eqtr4di 2814	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5	4	eleq2d 2847	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥 ∈ 𝐵))
6	4	rexeqdv 3320	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦 ↔ ∃𝑧 ∈ 𝐵 (𝑧(.r‘𝑟)𝑥) = 𝑦))
7	5, 6	anbi12d 641	. . . . . 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 6863	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
9		dvdsr.3	. . . . . . . . . . 11 ⊢ · = (.r‘𝑅)
10	8, 9	eqtr4di 2814	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
11	10	oveqd 7409	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑧(.r‘𝑟)𝑥) = (𝑧 · 𝑥))
12	11	eqeq1d 2763	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑧(.r‘𝑟)𝑥) = 𝑦 ↔ (𝑧 · 𝑥) = 𝑦))
13	12	rexbidv 3185	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∃𝑧 ∈ 𝐵 (𝑧(.r‘𝑟)𝑥) = 𝑦 ↔ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦))
14	13	anbi2d 639	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧(.r‘𝑟)𝑥) = 𝑦) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)))
15	7, 14	bitrd 281	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)))
16	15	opabbidv 5165	. . . 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 20385	. . . 4 ⊢ ∥r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦)})
18	3	fvexi 6877	. . . . 5 ⊢ 𝐵 ∈ V
19		eqcom 2768	. . . . . . . . 9 ⊢ ((𝑧 · 𝑥) = 𝑦 ↔ 𝑦 = (𝑧 · 𝑥))
20	19	rexbii 3108	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧 ∈ 𝐵 𝑦 = (𝑧 · 𝑥))
21	20	abbii 2828	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦} = {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑦 = (𝑧 · 𝑥)}
22	18	abrexex 7939	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑦 = (𝑧 · 𝑥)} ∈ V
23	21, 22	eqeltri 2857	. . . . . 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 7944	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} ∈ V
26	16, 17, 25	fvmpt 6971	. . 3 ⊢ (𝑅 ∈ V → (∥r‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)})
27	1, 26	eqtrid 2808	. 2 ⊢ (𝑅 ∈ V → ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)})
28		fvprc 6855	. . . 4 ⊢ (¬ 𝑅 ∈ V → (∥r‘𝑅) = ∅)
29	1, 28	eqtrid 2808	. . 3 ⊢ (¬ 𝑅 ∈ V → ∥ = ∅)
30		opabn0 5522	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} ≠ ∅ ↔ ∃𝑥∃𝑦(𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦))
31		n0i 4292	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → ¬ 𝐵 = ∅)
32		fvprc 6855	. . . . . . . . 9 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅)
33	3, 32	eqtrid 2808	. . . . . . . 8 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
34	31, 33	nsyl2 141	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → 𝑅 ∈ V)
35	34	adantr 484	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦) → 𝑅 ∈ V)
36	35	exlimivv 1951	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦) → 𝑅 ∈ V)
37	30, 36	sylbi 219	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} ≠ ∅ → 𝑅 ∈ V)
38	37	necon1bi 2984	. . 3 ⊢ (¬ 𝑅 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} = ∅)
39	29, 38	eqtr4d 2799	. 2 ⊢ (¬ 𝑅 ∈ V → ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)})
40	27, 39	pm2.61i 183	1 ⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)}

Syntax hints:  ¬ wn 3   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {cab 2739   ≠ wne 2956  ∃wrex 3085  Vcvv 3453  ∅c0 4285  {copab 5161  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  ._rcmulr 17270  ∥_rcdsr 20382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-ov 7395  df-dvdsr 20385

This theorem is referenced by:  dvdsr  20390  dvdsrpropd  20444  dvdsrzring  21493