Theorem dvdsrvald 13649

Description: Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
dvdsrvald.1	⊢ (𝜑 → 𝐵 = (Base‘𝑅))
dvdsrvald.2	⊢ (𝜑 → ∥ = (∥r‘𝑅))
dvdsrvald.r	⊢ (𝜑 → 𝑅 ∈ SRing)
dvdsrvald.3	⊢ (𝜑 → · = (.r‘𝑅))

Assertion

Ref	Expression
dvdsrvald	⊢ (𝜑 → ∥ = {⟨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   𝜑,𝑥,𝑦,𝑧

Allowed substitution hint:   ∥ (𝑧)

Proof of Theorem dvdsrvald

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dvdsr 13645	. . 3 ⊢ ∥r = (𝑟 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦)})
2		fveq2 5558	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3	2	eleq2d 2266	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↔ 𝑥 ∈ (Base‘𝑅)))
4		fveq2 5558	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
5	4	oveqd 5939	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑧(.r‘𝑟)𝑥) = (𝑧(.r‘𝑅)𝑥))
6	5	eqeq1d 2205	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝑧(.r‘𝑟)𝑥) = 𝑦 ↔ (𝑧(.r‘𝑅)𝑥) = 𝑦))
7	2, 6	rexeqbidv 2710	. . . . 5 ⊢ (𝑟 = 𝑅 → (∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦 ↔ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦))
8	3, 7	anbi12d 473	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)))
9	8	opabbidv 4099	. . 3 ⊢ (𝑟 = 𝑅 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑟) ∧ ∃𝑧 ∈ (Base‘𝑟)(𝑧(.r‘𝑟)𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)})
10		dvdsrvald.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ SRing)
11	10	elexd 2776	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
12		basfn 12736	. . . . . 6 ⊢ Base Fn V
13		funfvex 5575	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
14	13	funfni 5358	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
15	12, 11, 14	sylancr 414	. . . . 5 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
16		xpexg 4777	. . . . 5 ⊢ (((Base‘𝑅) ∈ V ∧ (Base‘𝑅) ∈ V) → ((Base‘𝑅) × (Base‘𝑅)) ∈ V)
17	15, 15, 16	syl2anc 411	. . . 4 ⊢ (𝜑 → ((Base‘𝑅) × (Base‘𝑅)) ∈ V)
18		simprr 531	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → (𝑧(.r‘𝑅)𝑥) = 𝑦)
19	10	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → 𝑅 ∈ SRing)
20		simprl 529	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → 𝑧 ∈ (Base‘𝑅))
21		simplr 528	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → 𝑥 ∈ (Base‘𝑅))
22		eqid 2196	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
23		eqid 2196	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
24	22, 23	srgcl 13526	. . . . . . . . . 10 ⊢ ((𝑅 ∈ SRing ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑧(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
25	19, 20, 21, 24	syl3anc 1249	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → (𝑧(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
26	18, 25	eqeltrrd 2274	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → 𝑦 ∈ (Base‘𝑅))
27	26	rexlimdvaa 2615	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦 → 𝑦 ∈ (Base‘𝑅)))
28	27	imdistanda 448	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦) → (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))))
29	28	ssopab2dv 4313	. . . . 5 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))})
30		df-xp 4669	. . . . 5 ⊢ ((Base‘𝑅) × (Base‘𝑅)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))}
31	29, 30	sseqtrrdi 3232	. . . 4 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ⊆ ((Base‘𝑅) × (Base‘𝑅)))
32	17, 31	ssexd 4173	. . 3 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ∈ V)
33	1, 9, 11, 32	fvmptd3 5655	. 2 ⊢ (𝜑 → (∥r‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)})
34		dvdsrvald.2	. 2 ⊢ (𝜑 → ∥ = (∥r‘𝑅))
35		dvdsrvald.1	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
36	35	eleq2d 2266	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝑅)))
37		dvdsrvald.3	. . . . . . 7 ⊢ (𝜑 → · = (.r‘𝑅))
38	37	oveqd 5939	. . . . . 6 ⊢ (𝜑 → (𝑧 · 𝑥) = (𝑧(.r‘𝑅)𝑥))
39	38	eqeq1d 2205	. . . . 5 ⊢ (𝜑 → ((𝑧 · 𝑥) = 𝑦 ↔ (𝑧(.r‘𝑅)𝑥) = 𝑦))
40	35, 39	rexeqbidv 2710	. . . 4 ⊢ (𝜑 → (∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦))
41	36, 40	anbi12d 473	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)))
42	41	opabbidv 4099	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)})
43	33, 34, 42	3eqtr4d 2239	1 ⊢ (𝜑 → ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  ∃wrex 2476  Vcvv 2763  {copab 4093   × cxp 4661   Fn wfn 5253  ‘cfv 5258  (class class class)co 5922  Basecbs 12678  ._rcmulr 12756  SRingcsrg 13519  ∥_rcdsr 13642

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-mgp 13477  df-srg 13520  df-dvdsr 13645

This theorem is referenced by:  dvdsrd  13650  dvdsrex  13654  dvdsrpropdg  13703  dvdsrzring  14159