Theorem unitpropdg 13647

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 13645	. . . . . 6 ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐿))
7	6	breq2d 4042	. . . . 5 ⊢ (𝜑 → (𝑧(∥r‘𝐾)(1r‘𝐾) ↔ 𝑧(∥r‘𝐾)(1r‘𝐿)))
8	6	breq2d 4042	. . . . 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 13546	. . . . . . . 8 ⊢ (𝐾 ∈ Ring → 𝐾 ∈ SRing)
11	4, 10	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ SRing)
12		ringsrg 13546	. . . . . . . 8 ⊢ (𝐿 ∈ Ring → 𝐿 ∈ SRing)
13	5, 12	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ SRing)
14	1, 2, 3, 11, 13	dvdsrpropdg 13646	. . . . . 6 ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐿))
15	14	breqd 4041	. . . . 5 ⊢ (𝜑 → (𝑧(∥r‘𝐾)(1r‘𝐿) ↔ 𝑧(∥r‘𝐿)(1r‘𝐿)))
16		eqid 2193	. . . . . . . . . 10 ⊢ (oppr‘𝐾) = (oppr‘𝐾)
17		eqid 2193	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
18	16, 17	opprbasg 13574	. . . . . . . . 9 ⊢ (𝐾 ∈ Ring → (Base‘𝐾) = (Base‘(oppr‘𝐾)))
19	4, 18	syl 14	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(oppr‘𝐾)))
20	1, 19	eqtrd 2226	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘(oppr‘𝐾)))
21		eqid 2193	. . . . . . . . . 10 ⊢ (oppr‘𝐿) = (oppr‘𝐿)
22		eqid 2193	. . . . . . . . . 10 ⊢ (Base‘𝐿) = (Base‘𝐿)
23	21, 22	opprbasg 13574	. . . . . . . . 9 ⊢ (𝐿 ∈ Ring → (Base‘𝐿) = (Base‘(oppr‘𝐿)))
24	5, 23	syl 14	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐿) = (Base‘(oppr‘𝐿)))
25	2, 24	eqtrd 2226	. . . . . . 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 2193	. . . . . . . . . 10 ⊢ (.r‘𝐾) = (.r‘𝐾)
31		eqid 2193	. . . . . . . . . 10 ⊢ (.r‘(oppr‘𝐾)) = (.r‘(oppr‘𝐾))
32	17, 30, 16, 31	opprmulg 13570	. . . . . . . . 9 ⊢ ((𝐾 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦(.r‘(oppr‘𝐾))𝑥) = (𝑥(.r‘𝐾)𝑦))
33	27, 28, 29, 32	syl3anc 1249	. . . . . . . 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 2193	. . . . . . . . . 10 ⊢ (.r‘𝐿) = (.r‘𝐿)
36		eqid 2193	. . . . . . . . . 10 ⊢ (.r‘(oppr‘𝐿)) = (.r‘(oppr‘𝐿))
37	22, 35, 21, 36	opprmulg 13570	. . . . . . . . 9 ⊢ ((𝐿 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦(.r‘(oppr‘𝐿))𝑥) = (𝑥(.r‘𝐿)𝑦))
38	34, 28, 29, 37	syl3anc 1249	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(.r‘(oppr‘𝐿))𝑥) = (𝑥(.r‘𝐿)𝑦))
39	26, 33, 38	3eqtr4d 2236	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(.r‘(oppr‘𝐾))𝑥) = (𝑦(.r‘(oppr‘𝐿))𝑥))
40	16	opprring 13578	. . . . . . . 8 ⊢ (𝐾 ∈ Ring → (oppr‘𝐾) ∈ Ring)
41		ringsrg 13546	. . . . . . . 8 ⊢ ((oppr‘𝐾) ∈ Ring → (oppr‘𝐾) ∈ SRing)
42	4, 40, 41	3syl 17	. . . . . . 7 ⊢ (𝜑 → (oppr‘𝐾) ∈ SRing)
43	21	opprring 13578	. . . . . . . 8 ⊢ (𝐿 ∈ Ring → (oppr‘𝐿) ∈ Ring)
44		ringsrg 13546	. . . . . . . 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 13646	. . . . . 6 ⊢ (𝜑 → (∥r‘(oppr‘𝐾)) = (∥r‘(oppr‘𝐿)))
47	46	breqd 4041	. . . . 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 2194	. . . 4 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐾))
51		eqidd 2194	. . . 4 ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐾))
52		eqidd 2194	. . . 4 ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐾))
53		eqidd 2194	. . . 4 ⊢ (𝜑 → (oppr‘𝐾) = (oppr‘𝐾))
54		eqidd 2194	. . . 4 ⊢ (𝜑 → (∥r‘(oppr‘𝐾)) = (∥r‘(oppr‘𝐾)))
55	50, 51, 52, 53, 54, 11	isunitd 13605	. . 3 ⊢ (𝜑 → (𝑧 ∈ (Unit‘𝐾) ↔ (𝑧(∥r‘𝐾)(1r‘𝐾) ∧ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐾))))
56		eqidd 2194	. . . 4 ⊢ (𝜑 → (Unit‘𝐿) = (Unit‘𝐿))
57		eqidd 2194	. . . 4 ⊢ (𝜑 → (1r‘𝐿) = (1r‘𝐿))
58		eqidd 2194	. . . 4 ⊢ (𝜑 → (∥r‘𝐿) = (∥r‘𝐿))
59		eqidd 2194	. . . 4 ⊢ (𝜑 → (oppr‘𝐿) = (oppr‘𝐿))
60		eqidd 2194	. . . 4 ⊢ (𝜑 → (∥r‘(oppr‘𝐿)) = (∥r‘(oppr‘𝐿)))
61	56, 57, 58, 59, 60, 13	isunitd 13605	. . 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 2191	1 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164   class class class wbr 4030  ‘cfv 5255  (class class class)co 5919  Basecbs 12621  ._rcmulr 12699  1_rcur 13458  SRingcsrg 13462  Ringcrg 13495  opp_rcoppr 13566  ∥_rcdsr 13585  Unitcui 13586

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-lttrn 7988  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-tpos 6300  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-plusg 12711  df-mulr 12712  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-minusg 13079  df-cmn 13359  df-abl 13360  df-mgp 13420  df-ur 13459  df-srg 13463  df-ring 13497  df-oppr 13567  df-dvdsr 13588  df-unit 13589

This theorem is referenced by:  invrpropdg  13648