Theorem dvdsrpropdg 13827

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
dvdsrpropdg.1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
dvdsrpropdg.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
dvdsrpropdg.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
dvdsrpropdg.k	⊢ (𝜑 → 𝐾 ∈ SRing)
dvdsrpropdg.l	⊢ (𝜑 → 𝐿 ∈ SRing)

Assertion

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

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

Proof of Theorem dvdsrpropdg

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvdsrpropdg.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
2	1	anassrs 400	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
3	2	eqeq1d 2213	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝑥(.r‘𝐾)𝑦) = 𝑧 ↔ (𝑥(.r‘𝐿)𝑦) = 𝑧))
4	3	an32s 568	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥(.r‘𝐾)𝑦) = 𝑧 ↔ (𝑥(.r‘𝐿)𝑦) = 𝑧))
5	4	rexbidva 2502	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐾)𝑦) = 𝑧 ↔ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐿)𝑦) = 𝑧))
6	5	pm5.32da 452	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐾)𝑦) = 𝑧) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐿)𝑦) = 𝑧)))
7		dvdsrpropdg.1	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
8	7	eleq2d 2274	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝐾)))
9	7	rexeqdv 2708	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐾)𝑦) = 𝑧 ↔ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r‘𝐾)𝑦) = 𝑧))
10	8, 9	anbi12d 473	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐾)𝑦) = 𝑧) ↔ (𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r‘𝐾)𝑦) = 𝑧)))
11		dvdsrpropdg.2	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
12	11	eleq2d 2274	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝐿)))
13	11	rexeqdv 2708	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝐿)𝑦) = 𝑧 ↔ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r‘𝐿)𝑦) = 𝑧))
14	12, 13	anbi12d 473	. . . 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 218	. . 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 4109	. 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		eqidd 2205	. . 3 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐾))
18		eqidd 2205	. . 3 ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐾))
19		dvdsrpropdg.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ SRing)
20		eqidd 2205	. . 3 ⊢ (𝜑 → (.r‘𝐾) = (.r‘𝐾))
21	17, 18, 19, 20	dvdsrvald 13773	. 2 ⊢ (𝜑 → (∥r‘𝐾) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(𝑥(.r‘𝐾)𝑦) = 𝑧)})
22		eqidd 2205	. . 3 ⊢ (𝜑 → (Base‘𝐿) = (Base‘𝐿))
23		eqidd 2205	. . 3 ⊢ (𝜑 → (∥r‘𝐿) = (∥r‘𝐿))
24		dvdsrpropdg.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ SRing)
25		eqidd 2205	. . 3 ⊢ (𝜑 → (.r‘𝐿) = (.r‘𝐿))
26	22, 23, 24, 25	dvdsrvald 13773	. 2 ⊢ (𝜑 → (∥r‘𝐿) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ (Base‘𝐿) ∧ ∃𝑥 ∈ (Base‘𝐿)(𝑥(.r‘𝐿)𝑦) = 𝑧)})
27	16, 21, 26	3eqtr4d 2247	1 ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1372   ∈ wcel 2175  ∃wrex 2484  {copab 4103  ‘cfv 5268  (class class class)co 5934  Basecbs 12751  ._rcmulr 12829  SRingcsrg 13643  ∥_rcdsr 13766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-cnex 7998  ax-resscn 7999  ax-1cn 8000  ax-1re 8001  ax-icn 8002  ax-addcl 8003  ax-addrcl 8004  ax-mulcl 8005  ax-addcom 8007  ax-addass 8009  ax-i2m1 8012  ax-0lt1 8013  ax-0id 8015  ax-rnegex 8016  ax-pre-ltirr 8019  ax-pre-ltadd 8023

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-iota 5229  df-fun 5270  df-fn 5271  df-fv 5276  df-riota 5889  df-ov 5937  df-oprab 5938  df-mpo 5939  df-pnf 8091  df-mnf 8092  df-ltxr 8094  df-inn 9019  df-2 9077  df-3 9078  df-ndx 12754  df-slot 12755  df-base 12757  df-sets 12758  df-plusg 12841  df-mulr 12842  df-0g 13008  df-mgm 13106  df-sgrp 13152  df-mnd 13167  df-mgp 13601  df-srg 13644  df-dvdsr 13769

This theorem is referenced by:  unitpropdg  13828