Theorem unitpropdg 14077

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

Hypotheses

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

Assertion

Ref	Expression
unitpropdg	⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿))

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

Proof of Theorem unitpropdg

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unitpropdg.1	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
2		unitpropdg.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
3		unitpropdg.3	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
4		unitpropdg.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Ring)
5		unitpropdg.l	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ Ring)
6	1, 2, 3, 4, 5	rngidpropdg 14075	. . . . . 6 ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐿))
7	6	breq2d 4074	. . . . 5 ⊢ (𝜑 → (𝑧(∥r‘𝐾)(1r‘𝐾) ↔ 𝑧(∥r‘𝐾)(1r‘𝐿)))
8	6	breq2d 4074	. . . . 5 ⊢ (𝜑 → (𝑧(∥r‘(oppr‘𝐾))(1r‘𝐾) ↔ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐿)))
9	7, 8	anbi12d 473	. . . 4 ⊢ (𝜑 → ((𝑧(∥r‘𝐾)(1r‘𝐾) ∧ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐾)) ↔ (𝑧(∥r‘𝐾)(1r‘𝐿) ∧ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐿))))
10		ringsrg 13976	. . . . . . . 8 ⊢ (𝐾 ∈ Ring → 𝐾 ∈ SRing)
11	4, 10	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ SRing)
12		ringsrg 13976	. . . . . . . 8 ⊢ (𝐿 ∈ Ring → 𝐿 ∈ SRing)
13	5, 12	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ SRing)
14	1, 2, 3, 11, 13	dvdsrpropdg 14076	. . . . . 6 ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐿))
15	14	breqd 4073	. . . . 5 ⊢ (𝜑 → (𝑧(∥r‘𝐾)(1r‘𝐿) ↔ 𝑧(∥r‘𝐿)(1r‘𝐿)))
16		eqid 2209	. . . . . . . . . 10 ⊢ (oppr‘𝐾) = (oppr‘𝐾)
17		eqid 2209	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
18	16, 17	opprbasg 14004	. . . . . . . . 9 ⊢ (𝐾 ∈ Ring → (Base‘𝐾) = (Base‘(oppr‘𝐾)))
19	4, 18	syl 14	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(oppr‘𝐾)))
20	1, 19	eqtrd 2242	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘(oppr‘𝐾)))
21		eqid 2209	. . . . . . . . . 10 ⊢ (oppr‘𝐿) = (oppr‘𝐿)
22		eqid 2209	. . . . . . . . . 10 ⊢ (Base‘𝐿) = (Base‘𝐿)
23	21, 22	opprbasg 14004	. . . . . . . . 9 ⊢ (𝐿 ∈ Ring → (Base‘𝐿) = (Base‘(oppr‘𝐿)))
24	5, 23	syl 14	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐿) = (Base‘(oppr‘𝐿)))
25	2, 24	eqtrd 2242	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘(oppr‘𝐿)))
26	3	ancom2s 566	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))
27	4	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝐾 ∈ Ring)
28		simprl 529	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
29		simprr 531	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
30		eqid 2209	. . . . . . . . . 10 ⊢ (.r‘𝐾) = (.r‘𝐾)
31		eqid 2209	. . . . . . . . . 10 ⊢ (.r‘(oppr‘𝐾)) = (.r‘(oppr‘𝐾))
32	17, 30, 16, 31	opprmulg 14000	. . . . . . . . 9 ⊢ ((𝐾 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦(.r‘(oppr‘𝐾))𝑥) = (𝑥(.r‘𝐾)𝑦))
33	27, 28, 29, 32	syl3anc 1252	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(.r‘(oppr‘𝐾))𝑥) = (𝑥(.r‘𝐾)𝑦))
34	5	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝐿 ∈ Ring)
35		eqid 2209	. . . . . . . . . 10 ⊢ (.r‘𝐿) = (.r‘𝐿)
36		eqid 2209	. . . . . . . . . 10 ⊢ (.r‘(oppr‘𝐿)) = (.r‘(oppr‘𝐿))
37	22, 35, 21, 36	opprmulg 14000	. . . . . . . . 9 ⊢ ((𝐿 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦(.r‘(oppr‘𝐿))𝑥) = (𝑥(.r‘𝐿)𝑦))
38	34, 28, 29, 37	syl3anc 1252	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(.r‘(oppr‘𝐿))𝑥) = (𝑥(.r‘𝐿)𝑦))
39	26, 33, 38	3eqtr4d 2252	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(.r‘(oppr‘𝐾))𝑥) = (𝑦(.r‘(oppr‘𝐿))𝑥))
40	16	opprring 14008	. . . . . . . 8 ⊢ (𝐾 ∈ Ring → (oppr‘𝐾) ∈ Ring)
41		ringsrg 13976	. . . . . . . 8 ⊢ ((oppr‘𝐾) ∈ Ring → (oppr‘𝐾) ∈ SRing)
42	4, 40, 41	3syl 17	. . . . . . 7 ⊢ (𝜑 → (oppr‘𝐾) ∈ SRing)
43	21	opprring 14008	. . . . . . . 8 ⊢ (𝐿 ∈ Ring → (oppr‘𝐿) ∈ Ring)
44		ringsrg 13976	. . . . . . . 8 ⊢ ((oppr‘𝐿) ∈ Ring → (oppr‘𝐿) ∈ SRing)
45	5, 43, 44	3syl 17	. . . . . . 7 ⊢ (𝜑 → (oppr‘𝐿) ∈ SRing)
46	20, 25, 39, 42, 45	dvdsrpropdg 14076	. . . . . 6 ⊢ (𝜑 → (∥r‘(oppr‘𝐾)) = (∥r‘(oppr‘𝐿)))
47	46	breqd 4073	. . . . 5 ⊢ (𝜑 → (𝑧(∥r‘(oppr‘𝐾))(1r‘𝐿) ↔ 𝑧(∥r‘(oppr‘𝐿))(1r‘𝐿)))
48	15, 47	anbi12d 473	. . . 4 ⊢ (𝜑 → ((𝑧(∥r‘𝐾)(1r‘𝐿) ∧ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐿)) ↔ (𝑧(∥r‘𝐿)(1r‘𝐿) ∧ 𝑧(∥r‘(oppr‘𝐿))(1r‘𝐿))))
49	9, 48	bitrd 188	. . 3 ⊢ (𝜑 → ((𝑧(∥r‘𝐾)(1r‘𝐾) ∧ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐾)) ↔ (𝑧(∥r‘𝐿)(1r‘𝐿) ∧ 𝑧(∥r‘(oppr‘𝐿))(1r‘𝐿))))
50		eqidd 2210	. . . 4 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐾))
51		eqidd 2210	. . . 4 ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐾))
52		eqidd 2210	. . . 4 ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐾))
53		eqidd 2210	. . . 4 ⊢ (𝜑 → (oppr‘𝐾) = (oppr‘𝐾))
54		eqidd 2210	. . . 4 ⊢ (𝜑 → (∥r‘(oppr‘𝐾)) = (∥r‘(oppr‘𝐾)))
55	50, 51, 52, 53, 54, 11	isunitd 14035	. . 3 ⊢ (𝜑 → (𝑧 ∈ (Unit‘𝐾) ↔ (𝑧(∥r‘𝐾)(1r‘𝐾) ∧ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐾))))
56		eqidd 2210	. . . 4 ⊢ (𝜑 → (Unit‘𝐿) = (Unit‘𝐿))
57		eqidd 2210	. . . 4 ⊢ (𝜑 → (1r‘𝐿) = (1r‘𝐿))
58		eqidd 2210	. . . 4 ⊢ (𝜑 → (∥r‘𝐿) = (∥r‘𝐿))
59		eqidd 2210	. . . 4 ⊢ (𝜑 → (oppr‘𝐿) = (oppr‘𝐿))
60		eqidd 2210	. . . 4 ⊢ (𝜑 → (∥r‘(oppr‘𝐿)) = (∥r‘(oppr‘𝐿)))
61	56, 57, 58, 59, 60, 13	isunitd 14035	. . 3 ⊢ (𝜑 → (𝑧 ∈ (Unit‘𝐿) ↔ (𝑧(∥r‘𝐿)(1r‘𝐿) ∧ 𝑧(∥r‘(oppr‘𝐿))(1r‘𝐿))))
62	49, 55, 61	3bitr4d 220	. 2 ⊢ (𝜑 → (𝑧 ∈ (Unit‘𝐾) ↔ 𝑧 ∈ (Unit‘𝐿)))
63	62	eqrdv 2207	1 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1375   ∈ wcel 2180   class class class wbr 4062  ‘cfv 5294  (class class class)co 5974  Basecbs 12998  ._rcmulr 13077  1_rcur 13888  SRingcsrg 13892  Ringcrg 13925  opp_rcoppr 13996  ∥_rcdsr 14015  Unitcui 14016

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-addass 8069  ax-i2m1 8072  ax-0lt1 8073  ax-0id 8075  ax-rnegex 8076  ax-pre-ltirr 8079  ax-pre-lttrn 8081  ax-pre-ltadd 8083

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-tpos 6361  df-pnf 8151  df-mnf 8152  df-ltxr 8154  df-inn 9079  df-2 9137  df-3 9138  df-ndx 13001  df-slot 13002  df-base 13004  df-sets 13005  df-plusg 13089  df-mulr 13090  df-0g 13257  df-mgm 13355  df-sgrp 13401  df-mnd 13416  df-grp 13502  df-minusg 13503  df-cmn 13789  df-abl 13790  df-mgp 13850  df-ur 13889  df-srg 13893  df-ring 13927  df-oppr 13997  df-dvdsr 14018  df-unit 14019

This theorem is referenced by:  invrpropdg  14078