Theorem subrgdv 14655

Description: A subring always has the same division function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)

Hypotheses

Ref	Expression
subrgdv.1	⊢ 𝑆 = (𝑅 ↾s 𝐴)
subrgdv.2	⊢ / = (/r‘𝑅)
subrgdv.3	⊢ 𝑈 = (Unit‘𝑆)
subrgdv.4	⊢ 𝐸 = (/r‘𝑆)

Assertion

Ref	Expression
subrgdv	⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋𝐸𝑌))

Proof of Theorem subrgdv

Step	Hyp	Ref	Expression
1		subrgdv.1	. . . . . 6 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
2		eqid 2238	. . . . . 6 ⊢ (invr‘𝑅) = (invr‘𝑅)
3		subrgdv.3	. . . . . 6 ⊢ 𝑈 = (Unit‘𝑆)
4		eqid 2238	. . . . . 6 ⊢ (invr‘𝑆) = (invr‘𝑆)
5	1, 2, 3, 4	subrginv 14654	. . . . 5 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑌 ∈ 𝑈) → ((invr‘𝑅)‘𝑌) = ((invr‘𝑆)‘𝑌))
6	5	3adant2 1047	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → ((invr‘𝑅)‘𝑌) = ((invr‘𝑆)‘𝑌))
7	6	oveq2d 6104	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋(.r‘𝑅)((invr‘𝑅)‘𝑌)) = (𝑋(.r‘𝑅)((invr‘𝑆)‘𝑌)))
8		subrgrcl 14643	. . . . . 6 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
9		eqid 2238	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
10	1, 9	ressmulrg 13558	. . . . . 6 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑅 ∈ Ring) → (.r‘𝑅) = (.r‘𝑆))
11	8, 10	mpdan 425	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (.r‘𝑅) = (.r‘𝑆))
12	11	3ad2ant1 1049	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (.r‘𝑅) = (.r‘𝑆))
13	12	oveqd 6105	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋(.r‘𝑅)((invr‘𝑆)‘𝑌)) = (𝑋(.r‘𝑆)((invr‘𝑆)‘𝑌)))
14	7, 13	eqtrd 2271	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋(.r‘𝑅)((invr‘𝑅)‘𝑌)) = (𝑋(.r‘𝑆)((invr‘𝑆)‘𝑌)))
15		eqidd 2239	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (Base‘𝑅) = (Base‘𝑅))
16		eqidd 2239	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (.r‘𝑅) = (.r‘𝑅))
17		eqidd 2239	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (Unit‘𝑅) = (Unit‘𝑅))
18		eqidd 2239	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (invr‘𝑅) = (invr‘𝑅))
19		subrgdv.2	. . . 4 ⊢ / = (/r‘𝑅)
20	19	a1i 9	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → / = (/r‘𝑅))
21	8	3ad2ant1 1049	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑅 ∈ Ring)
22		eqid 2238	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
23	22	subrgss 14639	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
24	23	3ad2ant1 1049	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝐴 ⊆ (Base‘𝑅))
25		simp2 1029	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ 𝐴)
26	24, 25	sseldd 3249	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑅))
27		eqid 2238	. . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
28	1, 27, 3	subrguss 14653	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑈 ⊆ (Unit‘𝑅))
29	28	3ad2ant1 1049	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑈 ⊆ (Unit‘𝑅))
30		simp3 1030	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ 𝑈)
31	29, 30	sseldd 3249	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑌 ∈ (Unit‘𝑅))
32	15, 16, 17, 18, 20, 21, 26, 31	dvrvald 14550	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋(.r‘𝑅)((invr‘𝑅)‘𝑌)))
33		eqidd 2239	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (Base‘𝑆) = (Base‘𝑆))
34		eqidd 2239	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (.r‘𝑆) = (.r‘𝑆))
35	3	a1i 9	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑈 = (Unit‘𝑆))
36		eqidd 2239	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (invr‘𝑆) = (invr‘𝑆))
37		subrgdv.4	. . . 4 ⊢ 𝐸 = (/r‘𝑆)
38	37	a1i 9	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝐸 = (/r‘𝑆))
39	1	subrgring 14641	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
40	39	3ad2ant1 1049	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑆 ∈ Ring)
41	1	subrgbas 14647	. . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
42	41	3ad2ant1 1049	. . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝐴 = (Base‘𝑆))
43	25, 42	eleqtrd 2317	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ (Base‘𝑆))
44	33, 34, 35, 36, 38, 40, 43, 30	dvrvald 14550	. 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋𝐸𝑌) = (𝑋(.r‘𝑆)((invr‘𝑆)‘𝑌)))
45	14, 32, 44	3eqtr4d 2281	1 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋𝐸𝑌))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209   ⊆ wss 3220  ‘cfv 5380  (class class class)co 6088  Basecbs 13410   ↾_s cress 13411  ._rcmulr 13491  Ringcrg 14407  Unitcui 14502  inv_rcinvr 14536  /_rcdvr 14547  SubRingcsubrg 14634

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-pre-ltirr 8297  ax-pre-lttrn 8299  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-tpos 6520  df-pnf 8368  df-mnf 8369  df-ltxr 8371  df-inn 9314  df-2 9372  df-3 9373  df-ndx 13413  df-slot 13414  df-base 13416  df-sets 13417  df-iress 13418  df-plusg 13503  df-mulr 13504  df-0g 13671  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-grp 13867  df-minusg 13868  df-subg 14032  df-cmn 14179  df-abl 14180  df-mgp 14325  df-ur 14370  df-srg 14375  df-ring 14409  df-oppr 14482  df-dvdsr 14504  df-unit 14505  df-invr 14537  df-dvr 14548  df-subrg 14636

This theorem is used by: (None)