Theorem dvdsr 20265

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 20263	. . 3 ⊢ Rel ∥
3	2	brrelex12i 5678	. 2 ⊢ (𝑋 ∥ 𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
4		elex 3459	. . 3 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ V)
5		id 22	. . . . 5 ⊢ ((𝑧 · 𝑋) = 𝑌 → (𝑧 · 𝑋) = 𝑌)
6		ovex 7386	. . . . 5 ⊢ (𝑧 · 𝑋) ∈ V
7	5, 6	eqeltrrdi 2837	. . . 4 ⊢ ((𝑧 · 𝑋) = 𝑌 → 𝑌 ∈ V)
8	7	rexlimivw 3126	. . 3 ⊢ (∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌 → 𝑌 ∈ V)
9	4, 8	anim12i 613	. 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 2813	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵))
12	10	oveq2d 7369	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑧 · 𝑥) = (𝑧 · 𝑋))
13		simpr 484	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
14	12, 13	eqeq12d 2745	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑧 · 𝑥) = 𝑦 ↔ (𝑧 · 𝑋) = 𝑌))
15	14	rexbidv 3153	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌))
16	11, 15	anbi12d 632	. . 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 20264	. . 3 ⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)}
20	16, 19	brabga 5481	. 2 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∥ 𝑌 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌)))
21	3, 9, 20	pm5.21nii 378	1 ⊢ (𝑋 ∥ 𝑌 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3438   class class class wbr 5095  ‘cfv 6486  (class class class)co 7353  Basecbs 17138  ._rcmulr 17180  ∥_rcdsr 20257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-dvdsr 20260

This theorem is referenced by:  dvdsr2  20266  dvdsrmul  20267  dvdsrcl  20268  dvdsrcl2  20269  dvdsrtr  20271  dvdsrmul1  20272  opprunit  20280  crngunit  20281  rhmdvdsr  20411  subrgdvds  20489  isunit2  33190  dvdsruassoi  33331  dvdsruasso  33332  dvdsrspss  33334  rprmasso2  33473  unitmulrprm  33475  rprmirredlem  33477  1arithufdlem3  33493  rhmqusspan  42158  unitscyglem5  42172