Theorem dvdsrpropd 20332

Description: The divisibility relation depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)

Hypotheses

Ref	Expression
rngidpropd.1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
rngidpropd.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
rngidpropd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))

Assertion

Ref	Expression
dvdsrpropd	⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐿))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Proof of Theorem dvdsrpropd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngidpropd.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
2	1	anassrs 467	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
3	2	eqeq1d 2732	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝑥(.r‘𝐾)𝑦) = 𝑧 ↔ (𝑥(.r‘𝐿)𝑦) = 𝑧))
4	3	an32s 652	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥(.r‘𝐾)𝑦) = 𝑧 ↔ (𝑥(.r‘𝐿)𝑦) = 𝑧))
5	4	rexbidva 3156	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐾)𝑦) = 𝑧 ↔ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐿)𝑦) = 𝑧))
6	5	pm5.32da 579	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐾)𝑦) = 𝑧) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐿)𝑦) = 𝑧)))
7		rngidpropd.1	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
8	7	eleq2d 2815	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝐾)))
9	7	rexeqdv 3302	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐾)𝑦) = 𝑧 ↔ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r‘𝐾)𝑦) = 𝑧))
10	8, 9	anbi12d 632	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐾)𝑦) = 𝑧) ↔ (𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r‘𝐾)𝑦) = 𝑧)))
11		rngidpropd.2	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
12	11	eleq2d 2815	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝐿)))
13	11	rexeqdv 3302	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐿)𝑦) = 𝑧 ↔ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r‘𝐿)𝑦) = 𝑧))
14	12, 13	anbi12d 632	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐿)𝑦) = 𝑧) ↔ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r‘𝐿)𝑦) = 𝑧)))
15	6, 10, 14	3bitr3d 309	. . 3 ⊢ (𝜑 → ((𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r‘𝐾)𝑦) = 𝑧) ↔ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r‘𝐿)𝑦) = 𝑧)))
16	15	opabbidv 5176	. 2 ⊢ (𝜑 → {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r‘𝐾)𝑦) = 𝑧)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r‘𝐿)𝑦) = 𝑧)})
17		eqid 2730	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2730	. . 3 ⊢ (∥r‘𝐾) = (∥r‘𝐾)
19		eqid 2730	. . 3 ⊢ (.r‘𝐾) = (.r‘𝐾)
20	17, 18, 19	dvdsrval 20277	. 2 ⊢ (∥r‘𝐾) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r‘𝐾)𝑦) = 𝑧)}
21		eqid 2730	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
22		eqid 2730	. . 3 ⊢ (∥r‘𝐿) = (∥r‘𝐿)
23		eqid 2730	. . 3 ⊢ (.r‘𝐿) = (.r‘𝐿)
24	21, 22, 23	dvdsrval 20277	. 2 ⊢ (∥r‘𝐿) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r‘𝐿)𝑦) = 𝑧)}
25	16, 20, 24	3eqtr4g 2790	1 ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐿))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3054  {copab 5172  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  ._rcmulr 17228  ∥_rcdsr 20270

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-dvdsr 20273

This theorem is referenced by:  unitpropd  20333