Theorem dvdsr 19803

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

Hypotheses

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

Assertion

Ref	Expression
dvdsr	⊢ (𝑋 ∥ 𝑌 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌))

Distinct variable groups:   𝑧,𝐵   𝑧,𝑋   𝑧,𝑌   𝑧,𝑅   𝑧, ·

Allowed substitution hint:   ∥ (𝑧)

Proof of Theorem dvdsr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvdsr.2	. . . 4 ⊢ ∥ = (∥r‘𝑅)
2	1	reldvdsr 19801	. . 3 ⊢ Rel ∥
3	2	brrelex12i 5633	. 2 ⊢ (𝑋 ∥ 𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
4		elex 3440	. . 3 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ V)
5		id 22	. . . . 5 ⊢ ((𝑧 · 𝑋) = 𝑌 → (𝑧 · 𝑋) = 𝑌)
6		ovex 7288	. . . . 5 ⊢ (𝑧 · 𝑋) ∈ V
7	5, 6	eqeltrrdi 2848	. . . 4 ⊢ ((𝑧 · 𝑋) = 𝑌 → 𝑌 ∈ V)
8	7	rexlimivw 3210	. . 3 ⊢ (∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌 → 𝑌 ∈ V)
9	4, 8	anim12i 612	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
10		simpl 482	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
11	10	eleq1d 2823	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵))
12	10	oveq2d 7271	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑧 · 𝑥) = (𝑧 · 𝑋))
13		simpr 484	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
14	12, 13	eqeq12d 2754	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑧 · 𝑥) = 𝑦 ↔ (𝑧 · 𝑋) = 𝑌))
15	14	rexbidv 3225	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌))
16	11, 15	anbi12d 630	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌)))
17		dvdsr.1	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
18		dvdsr.3	. . . 4 ⊢ · = (.r‘𝑅)
19	17, 1, 18	dvdsrval 19802	. . 3 ⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)}
20	16, 19	brabga 5440	. 2 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∥ 𝑌 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌)))
21	3, 9, 20	pm5.21nii 379	1 ⊢ (𝑋 ∥ 𝑌 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  Vcvv 3422   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  ._rcmulr 16889  ∥_rcdsr 19795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-dvdsr 19798

This theorem is referenced by:  dvdsr2  19804  dvdsrmul  19805  dvdsrcl  19806  dvdsrcl2  19807  dvdsrtr  19809  dvdsrmul1  19810  opprunit  19818  crngunit  19819  subrgdvds  19953  rhmdvdsr  31419