Theorem unitpropdg 13317

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 13315	. . . . . 6 ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐿))
7	6	breq2d 4016	. . . . 5 ⊢ (𝜑 → (𝑧(∥r‘𝐾)(1r‘𝐾) ↔ 𝑧(∥r‘𝐾)(1r‘𝐿)))
8	6	breq2d 4016	. . . . 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 13224	. . . . . . . 8 ⊢ (𝐾 ∈ Ring → 𝐾 ∈ SRing)
11	4, 10	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ SRing)
12		ringsrg 13224	. . . . . . . 8 ⊢ (𝐿 ∈ Ring → 𝐿 ∈ SRing)
13	5, 12	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ SRing)
14	1, 2, 3, 11, 13	dvdsrpropdg 13316	. . . . . 6 ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐿))
15	14	breqd 4015	. . . . 5 ⊢ (𝜑 → (𝑧(∥r‘𝐾)(1r‘𝐿) ↔ 𝑧(∥r‘𝐿)(1r‘𝐿)))
16		eqid 2177	. . . . . . . . . 10 ⊢ (oppr‘𝐾) = (oppr‘𝐾)
17		eqid 2177	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
18	16, 17	opprbasg 13247	. . . . . . . . 9 ⊢ (𝐾 ∈ Ring → (Base‘𝐾) = (Base‘(oppr‘𝐾)))
19	4, 18	syl 14	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(oppr‘𝐾)))
20	1, 19	eqtrd 2210	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘(oppr‘𝐾)))
21		eqid 2177	. . . . . . . . . 10 ⊢ (oppr‘𝐿) = (oppr‘𝐿)
22		eqid 2177	. . . . . . . . . 10 ⊢ (Base‘𝐿) = (Base‘𝐿)
23	21, 22	opprbasg 13247	. . . . . . . . 9 ⊢ (𝐿 ∈ Ring → (Base‘𝐿) = (Base‘(oppr‘𝐿)))
24	5, 23	syl 14	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐿) = (Base‘(oppr‘𝐿)))
25	2, 24	eqtrd 2210	. . . . . . 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 2177	. . . . . . . . . 10 ⊢ (.r‘𝐾) = (.r‘𝐾)
31		eqid 2177	. . . . . . . . . 10 ⊢ (.r‘(oppr‘𝐾)) = (.r‘(oppr‘𝐾))
32	17, 30, 16, 31	opprmulg 13243	. . . . . . . . 9 ⊢ ((𝐾 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦(.r‘(oppr‘𝐾))𝑥) = (𝑥(.r‘𝐾)𝑦))
33	27, 28, 29, 32	syl3anc 1238	. . . . . . . 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 2177	. . . . . . . . . 10 ⊢ (.r‘𝐿) = (.r‘𝐿)
36		eqid 2177	. . . . . . . . . 10 ⊢ (.r‘(oppr‘𝐿)) = (.r‘(oppr‘𝐿))
37	22, 35, 21, 36	opprmulg 13243	. . . . . . . . 9 ⊢ ((𝐿 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦(.r‘(oppr‘𝐿))𝑥) = (𝑥(.r‘𝐿)𝑦))
38	34, 28, 29, 37	syl3anc 1238	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(.r‘(oppr‘𝐿))𝑥) = (𝑥(.r‘𝐿)𝑦))
39	26, 33, 38	3eqtr4d 2220	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(.r‘(oppr‘𝐾))𝑥) = (𝑦(.r‘(oppr‘𝐿))𝑥))
40	16	opprring 13249	. . . . . . . 8 ⊢ (𝐾 ∈ Ring → (oppr‘𝐾) ∈ Ring)
41		ringsrg 13224	. . . . . . . 8 ⊢ ((oppr‘𝐾) ∈ Ring → (oppr‘𝐾) ∈ SRing)
42	4, 40, 41	3syl 17	. . . . . . 7 ⊢ (𝜑 → (oppr‘𝐾) ∈ SRing)
43	21	opprring 13249	. . . . . . . 8 ⊢ (𝐿 ∈ Ring → (oppr‘𝐿) ∈ Ring)
44		ringsrg 13224	. . . . . . . 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 13316	. . . . . 6 ⊢ (𝜑 → (∥r‘(oppr‘𝐾)) = (∥r‘(oppr‘𝐿)))
47	46	breqd 4015	. . . . 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 2178	. . . 4 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐾))
51		eqidd 2178	. . . 4 ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐾))
52		eqidd 2178	. . . 4 ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐾))
53		eqidd 2178	. . . 4 ⊢ (𝜑 → (oppr‘𝐾) = (oppr‘𝐾))
54		eqidd 2178	. . . 4 ⊢ (𝜑 → (∥r‘(oppr‘𝐾)) = (∥r‘(oppr‘𝐾)))
55	50, 51, 52, 53, 54, 11	isunitd 13275	. . 3 ⊢ (𝜑 → (𝑧 ∈ (Unit‘𝐾) ↔ (𝑧(∥r‘𝐾)(1r‘𝐾) ∧ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐾))))
56		eqidd 2178	. . . 4 ⊢ (𝜑 → (Unit‘𝐿) = (Unit‘𝐿))
57		eqidd 2178	. . . 4 ⊢ (𝜑 → (1r‘𝐿) = (1r‘𝐿))
58		eqidd 2178	. . . 4 ⊢ (𝜑 → (∥r‘𝐿) = (∥r‘𝐿))
59		eqidd 2178	. . . 4 ⊢ (𝜑 → (oppr‘𝐿) = (oppr‘𝐿))
60		eqidd 2178	. . . 4 ⊢ (𝜑 → (∥r‘(oppr‘𝐿)) = (∥r‘(oppr‘𝐿)))
61	56, 57, 58, 59, 60, 13	isunitd 13275	. . 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 2175	1 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148   class class class wbr 4004  ‘cfv 5217  (class class class)co 5875  Basecbs 12462  ._rcmulr 12537  1_rcur 13142  SRingcsrg 13146  Ringcrg 13179  opp_rcoppr 13239  ∥_rcdsr 13255  Unitcui 13256

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-pre-ltirr 7923  ax-pre-lttrn 7925  ax-pre-ltadd 7927

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-tpos 6246  df-pnf 7994  df-mnf 7995  df-ltxr 7997  df-inn 8920  df-2 8978  df-3 8979  df-ndx 12465  df-slot 12466  df-base 12468  df-sets 12469  df-plusg 12549  df-mulr 12550  df-0g 12707  df-mgm 12775  df-sgrp 12808  df-mnd 12818  df-grp 12880  df-minusg 12881  df-cmn 13090  df-abl 13091  df-mgp 13131  df-ur 13143  df-srg 13147  df-ring 13181  df-oppr 13240  df-dvdsr 13258  df-unit 13259

This theorem is referenced by:  invrpropdg  13318