Theorem dvdsrex 14265

Description: Existence of the divisibility relation. (Contributed by Jim Kingdon, 28-Jan-2025.)

Assertion

Ref	Expression
dvdsrex	⊢ (𝑅 ∈ SRing → (∥r‘𝑅) ∈ V)

Proof of Theorem dvdsrex

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

Step	Hyp	Ref	Expression
1		eqidd 2235	. . 3 ⊢ (𝑅 ∈ SRing → (Base‘𝑅) = (Base‘𝑅))
2		eqidd 2235	. . 3 ⊢ (𝑅 ∈ SRing → (∥r‘𝑅) = (∥r‘𝑅))
3		id 19	. . 3 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ SRing)
4		eqidd 2235	. . 3 ⊢ (𝑅 ∈ SRing → (.r‘𝑅) = (.r‘𝑅))
5	1, 2, 3, 4	dvdsrvald 14260	. 2 ⊢ (𝑅 ∈ SRing → (∥r‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)})
6		basfn 13292	. . . . 5 ⊢ Base Fn V
7		elex 2827	. . . . 5 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ V)
8		funfvex 5689	. . . . . 6 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
9	8	funfni 5460	. . . . 5 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
10	6, 7, 9	sylancr 414	. . . 4 ⊢ (𝑅 ∈ SRing → (Base‘𝑅) ∈ V)
11		xpexg 4866	. . . 4 ⊢ (((Base‘𝑅) ∈ V ∧ (Base‘𝑅) ∈ V) → ((Base‘𝑅) × (Base‘𝑅)) ∈ V)
12	10, 10, 11	syl2anc 411	. . 3 ⊢ (𝑅 ∈ SRing → ((Base‘𝑅) × (Base‘𝑅)) ∈ V)
13		simprr 533	. . . . . . . 8 ⊢ (((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → (𝑧(.r‘𝑅)𝑥) = 𝑦)
14		simpll 527	. . . . . . . . 9 ⊢ (((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → 𝑅 ∈ SRing)
15		simprl 531	. . . . . . . . 9 ⊢ (((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → 𝑧 ∈ (Base‘𝑅))
16		simplr 529	. . . . . . . . 9 ⊢ (((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → 𝑥 ∈ (Base‘𝑅))
17		eqid 2234	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
18		eqid 2234	. . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅)
19	17, 18	srgcl 14135	. . . . . . . . 9 ⊢ ((𝑅 ∈ SRing ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑧(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
20	14, 15, 16, 19	syl3anc 1274	. . . . . . . 8 ⊢ (((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → (𝑧(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
21	13, 20	eqeltrrd 2312	. . . . . . 7 ⊢ (((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ (𝑧(.r‘𝑅)𝑥) = 𝑦)) → 𝑦 ∈ (Base‘𝑅))
22	21	rexlimdvaa 2663	. . . . . 6 ⊢ ((𝑅 ∈ SRing ∧ 𝑥 ∈ (Base‘𝑅)) → (∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦 → 𝑦 ∈ (Base‘𝑅)))
23	22	imdistanda 448	. . . . 5 ⊢ (𝑅 ∈ SRing → ((𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦) → (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))))
24	23	ssopab2dv 4399	. . . 4 ⊢ (𝑅 ∈ SRing → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))})
25		df-xp 4757	. . . 4 ⊢ ((Base‘𝑅) × (Base‘𝑅)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))}
26	24, 25	sseqtrrdi 3289	. . 3 ⊢ (𝑅 ∈ SRing → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ⊆ ((Base‘𝑅) × (Base‘𝑅)))
27	12, 26	ssexd 4252	. 2 ⊢ (𝑅 ∈ SRing → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑧 ∈ (Base‘𝑅)(𝑧(.r‘𝑅)𝑥) = 𝑦)} ∈ V)
28	5, 27	eqeltrd 2311	1 ⊢ (𝑅 ∈ SRing → (∥r‘𝑅) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  ∃wrex 2523  Vcvv 2815  {copab 4172   × cxp 4749   Fn wfn 5349  ‘cfv 5354  (class class class)co 6052  Basecbs 13233  ._rcmulr 13312  SRingcsrg 14128  ∥_rcdsr 14252

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-addass 8234  ax-i2m1 8237  ax-0lt1 8238  ax-0id 8240  ax-rnegex 8241  ax-pre-ltirr 8244  ax-pre-ltadd 8248

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8315  df-mnf 8316  df-ltxr 8318  df-inn 9243  df-2 9301  df-3 9302  df-ndx 13236  df-slot 13237  df-base 13239  df-sets 13240  df-plusg 13324  df-mulr 13325  df-0g 13492  df-mgm 13590  df-sgrp 13636  df-mnd 13651  df-mgp 14086  df-srg 14129  df-dvdsr 14255

This theorem is referenced by:  isunitd  14273