Theorem dvdsrval 20434

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 6871	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3		dvdsr.1	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	eqtr4di 2818	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5	4	eleq2d 2851	. . . . . . 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 643	. . . . . 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 6871	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
9		dvdsr.3	. . . . . . . . . . 11 ⊢ · = (.r‘𝑅)
10	8, 9	eqtr4di 2818	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
11	10	oveqd 7417	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑧(.r‘𝑟)𝑥) = (𝑧 · 𝑥))
12	11	eqeq1d 2767	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑧(.r‘𝑟)𝑥) = 𝑦 ↔ (𝑧 · 𝑥) = 𝑦))
13	12	rexbidv 3189	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∃𝑧 ∈ 𝐵 (𝑧(.r‘𝑟)𝑥) = 𝑦 ↔ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦))
14	13	anbi2d 641	. . . . . 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 5171	. . . 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 20430	. . . 4 ⊢ ∥r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦)})
18	3	fvexi 6885	. . . . 5 ⊢ 𝐵 ∈ V
19		eqcom 2772	. . . . . . . . 9 ⊢ ((𝑧 · 𝑥) = 𝑦 ↔ 𝑦 = (𝑧 · 𝑥))
20	19	rexbii 3112	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧 ∈ 𝐵 𝑦 = (𝑧 · 𝑥))
21	20	abbii 2832	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦} = {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑦 = (𝑧 · 𝑥)}
22	18	abrexex 7947	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑧 ∈ 𝐵 𝑦 = (𝑧 · 𝑥)} ∈ V
23	21, 22	eqeltri 2861	. . . . . 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 7952	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} ∈ V
26	16, 17, 25	fvmpt 6979	. . 3 ⊢ (𝑅 ∈ V → (∥r‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)})
27	1, 26	eqtrid 2812	. 2 ⊢ (𝑅 ∈ V → ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)})
28		fvprc 6863	. . . 4 ⊢ (¬ 𝑅 ∈ V → (∥r‘𝑅) = ∅)
29	1, 28	eqtrid 2812	. . 3 ⊢ (¬ 𝑅 ∈ V → ∥ = ∅)
30		opabn0 5529	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} ≠ ∅ ↔ ∃𝑥∃𝑦(𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦))
31		n0i 4295	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → ¬ 𝐵 = ∅)
32		fvprc 6863	. . . . . . . . 9 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅)
33	3, 32	eqtrid 2812	. . . . . . . 8 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
34	31, 33	nsyl2 142	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → 𝑅 ∈ V)
35	34	adantr 485	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦) → 𝑅 ∈ V)
36	35	exlimivv 1955	. . . . 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 2988	. . 3 ⊢ (¬ 𝑅 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} = ∅)
39	29, 38	eqtr4d 2803	. 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)}

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145  {cab 2743   ≠ wne 2960  ∃wrex 3089  Vcvv 3457  ∅c0 4288  {copab 5167  ‘cfv 6525  (class class class)co 7400  Basecbs 17259  ._rcmulr 17301  ∥_rcdsr 20427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-dvdsr 20430

This theorem is referenced by:  dvdsr  20435  dvdsrpropd  20489  dvdsrzring  21571