Theorem unitpropdg 14120

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 14118	. . . . . 6 ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐿))
7	6	breq2d 4095	. . . . 5 ⊢ (𝜑 → (𝑧(∥r‘𝐾)(1r‘𝐾) ↔ 𝑧(∥r‘𝐾)(1r‘𝐿)))
8	6	breq2d 4095	. . . . 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 14018	. . . . . . . 8 ⊢ (𝐾 ∈ Ring → 𝐾 ∈ SRing)
11	4, 10	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ SRing)
12		ringsrg 14018	. . . . . . . 8 ⊢ (𝐿 ∈ Ring → 𝐿 ∈ SRing)
13	5, 12	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ SRing)
14	1, 2, 3, 11, 13	dvdsrpropdg 14119	. . . . . 6 ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐿))
15	14	breqd 4094	. . . . 5 ⊢ (𝜑 → (𝑧(∥r‘𝐾)(1r‘𝐿) ↔ 𝑧(∥r‘𝐿)(1r‘𝐿)))
16		eqid 2229	. . . . . . . . . 10 ⊢ (oppr‘𝐾) = (oppr‘𝐾)
17		eqid 2229	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
18	16, 17	opprbasg 14046	. . . . . . . . 9 ⊢ (𝐾 ∈ Ring → (Base‘𝐾) = (Base‘(oppr‘𝐾)))
19	4, 18	syl 14	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(oppr‘𝐾)))
20	1, 19	eqtrd 2262	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘(oppr‘𝐾)))
21		eqid 2229	. . . . . . . . . 10 ⊢ (oppr‘𝐿) = (oppr‘𝐿)
22		eqid 2229	. . . . . . . . . 10 ⊢ (Base‘𝐿) = (Base‘𝐿)
23	21, 22	opprbasg 14046	. . . . . . . . 9 ⊢ (𝐿 ∈ Ring → (Base‘𝐿) = (Base‘(oppr‘𝐿)))
24	5, 23	syl 14	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐿) = (Base‘(oppr‘𝐿)))
25	2, 24	eqtrd 2262	. . . . . . 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 2229	. . . . . . . . . 10 ⊢ (.r‘𝐾) = (.r‘𝐾)
31		eqid 2229	. . . . . . . . . 10 ⊢ (.r‘(oppr‘𝐾)) = (.r‘(oppr‘𝐾))
32	17, 30, 16, 31	opprmulg 14042	. . . . . . . . 9 ⊢ ((𝐾 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦(.r‘(oppr‘𝐾))𝑥) = (𝑥(.r‘𝐾)𝑦))
33	27, 28, 29, 32	syl3anc 1271	. . . . . . . 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 2229	. . . . . . . . . 10 ⊢ (.r‘𝐿) = (.r‘𝐿)
36		eqid 2229	. . . . . . . . . 10 ⊢ (.r‘(oppr‘𝐿)) = (.r‘(oppr‘𝐿))
37	22, 35, 21, 36	opprmulg 14042	. . . . . . . . 9 ⊢ ((𝐿 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦(.r‘(oppr‘𝐿))𝑥) = (𝑥(.r‘𝐿)𝑦))
38	34, 28, 29, 37	syl3anc 1271	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(.r‘(oppr‘𝐿))𝑥) = (𝑥(.r‘𝐿)𝑦))
39	26, 33, 38	3eqtr4d 2272	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(.r‘(oppr‘𝐾))𝑥) = (𝑦(.r‘(oppr‘𝐿))𝑥))
40	16	opprring 14050	. . . . . . . 8 ⊢ (𝐾 ∈ Ring → (oppr‘𝐾) ∈ Ring)
41		ringsrg 14018	. . . . . . . 8 ⊢ ((oppr‘𝐾) ∈ Ring → (oppr‘𝐾) ∈ SRing)
42	4, 40, 41	3syl 17	. . . . . . 7 ⊢ (𝜑 → (oppr‘𝐾) ∈ SRing)
43	21	opprring 14050	. . . . . . . 8 ⊢ (𝐿 ∈ Ring → (oppr‘𝐿) ∈ Ring)
44		ringsrg 14018	. . . . . . . 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 14119	. . . . . 6 ⊢ (𝜑 → (∥r‘(oppr‘𝐾)) = (∥r‘(oppr‘𝐿)))
47	46	breqd 4094	. . . . 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 2230	. . . 4 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐾))
51		eqidd 2230	. . . 4 ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐾))
52		eqidd 2230	. . . 4 ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐾))
53		eqidd 2230	. . . 4 ⊢ (𝜑 → (oppr‘𝐾) = (oppr‘𝐾))
54		eqidd 2230	. . . 4 ⊢ (𝜑 → (∥r‘(oppr‘𝐾)) = (∥r‘(oppr‘𝐾)))
55	50, 51, 52, 53, 54, 11	isunitd 14078	. . 3 ⊢ (𝜑 → (𝑧 ∈ (Unit‘𝐾) ↔ (𝑧(∥r‘𝐾)(1r‘𝐾) ∧ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐾))))
56		eqidd 2230	. . . 4 ⊢ (𝜑 → (Unit‘𝐿) = (Unit‘𝐿))
57		eqidd 2230	. . . 4 ⊢ (𝜑 → (1r‘𝐿) = (1r‘𝐿))
58		eqidd 2230	. . . 4 ⊢ (𝜑 → (∥r‘𝐿) = (∥r‘𝐿))
59		eqidd 2230	. . . 4 ⊢ (𝜑 → (oppr‘𝐿) = (oppr‘𝐿))
60		eqidd 2230	. . . 4 ⊢ (𝜑 → (∥r‘(oppr‘𝐿)) = (∥r‘(oppr‘𝐿)))
61	56, 57, 58, 59, 60, 13	isunitd 14078	. . 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 2227	1 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200   class class class wbr 4083  ‘cfv 5318  (class class class)co 6007  Basecbs 13040  ._rcmulr 13119  1_rcur 13930  SRingcsrg 13934  Ringcrg 13967  opp_rcoppr 14038  ∥_rcdsr 14057  Unitcui 14058

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-addass 8109  ax-i2m1 8112  ax-0lt1 8113  ax-0id 8115  ax-rnegex 8116  ax-pre-ltirr 8119  ax-pre-lttrn 8121  ax-pre-ltadd 8123

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-tpos 6397  df-pnf 8191  df-mnf 8192  df-ltxr 8194  df-inn 9119  df-2 9177  df-3 9178  df-ndx 13043  df-slot 13044  df-base 13046  df-sets 13047  df-plusg 13131  df-mulr 13132  df-0g 13299  df-mgm 13397  df-sgrp 13443  df-mnd 13458  df-grp 13544  df-minusg 13545  df-cmn 13831  df-abl 13832  df-mgp 13892  df-ur 13931  df-srg 13935  df-ring 13969  df-oppr 14039  df-dvdsr 14060  df-unit 14061

This theorem is referenced by:  invrpropdg  14121