Theorem unitpropdg 14504

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 14502	. . . . . 6 ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐿))
7	6	breq2d 4142	. . . . 5 ⊢ (𝜑 → (𝑧(∥r‘𝐾)(1r‘𝐾) ↔ 𝑧(∥r‘𝐾)(1r‘𝐿)))
8	6	breq2d 4142	. . . . 5 ⊢ (𝜑 → (𝑧(∥r‘(oppr‘𝐾))(1r‘𝐾) ↔ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐿)))
9	7, 8	anbi12d 477	. . . 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 14401	. . . . . . . 8 ⊢ (𝐾 ∈ Ring → 𝐾 ∈ SRing)
11	4, 10	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ SRing)
12		ringsrg 14401	. . . . . . . 8 ⊢ (𝐿 ∈ Ring → 𝐿 ∈ SRing)
13	5, 12	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ SRing)
14	1, 2, 3, 11, 13	dvdsrpropdg 14503	. . . . . 6 ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐿))
15	14	breqd 4141	. . . . 5 ⊢ (𝜑 → (𝑧(∥r‘𝐾)(1r‘𝐿) ↔ 𝑧(∥r‘𝐿)(1r‘𝐿)))
16		eqid 2238	. . . . . . . . . 10 ⊢ (oppr‘𝐾) = (oppr‘𝐾)
17		eqid 2238	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
18	16, 17	opprbasg 14429	. . . . . . . . 9 ⊢ (𝐾 ∈ Ring → (Base‘𝐾) = (Base‘(oppr‘𝐾)))
19	4, 18	syl 14	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(oppr‘𝐾)))
20	1, 19	eqtrd 2271	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘(oppr‘𝐾)))
21		eqid 2238	. . . . . . . . . 10 ⊢ (oppr‘𝐿) = (oppr‘𝐿)
22		eqid 2238	. . . . . . . . . 10 ⊢ (Base‘𝐿) = (Base‘𝐿)
23	21, 22	opprbasg 14429	. . . . . . . . 9 ⊢ (𝐿 ∈ Ring → (Base‘𝐿) = (Base‘(oppr‘𝐿)))
24	5, 23	syl 14	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐿) = (Base‘(oppr‘𝐿)))
25	2, 24	eqtrd 2271	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘(oppr‘𝐿)))
26	3	ancom2s 572	. . . . . . . 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 535	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
29		simprr 537	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
30		eqid 2238	. . . . . . . . . 10 ⊢ (.r‘𝐾) = (.r‘𝐾)
31		eqid 2238	. . . . . . . . . 10 ⊢ (.r‘(oppr‘𝐾)) = (.r‘(oppr‘𝐾))
32	17, 30, 16, 31	opprmulg 14425	. . . . . . . . 9 ⊢ ((𝐾 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦(.r‘(oppr‘𝐾))𝑥) = (𝑥(.r‘𝐾)𝑦))
33	27, 28, 29, 32	syl3anc 1278	. . . . . . . 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 2238	. . . . . . . . . 10 ⊢ (.r‘𝐿) = (.r‘𝐿)
36		eqid 2238	. . . . . . . . . 10 ⊢ (.r‘(oppr‘𝐿)) = (.r‘(oppr‘𝐿))
37	22, 35, 21, 36	opprmulg 14425	. . . . . . . . 9 ⊢ ((𝐿 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦(.r‘(oppr‘𝐿))𝑥) = (𝑥(.r‘𝐿)𝑦))
38	34, 28, 29, 37	syl3anc 1278	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(.r‘(oppr‘𝐿))𝑥) = (𝑥(.r‘𝐿)𝑦))
39	26, 33, 38	3eqtr4d 2281	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵)) → (𝑦(.r‘(oppr‘𝐾))𝑥) = (𝑦(.r‘(oppr‘𝐿))𝑥))
40	16	opprring 14433	. . . . . . . 8 ⊢ (𝐾 ∈ Ring → (oppr‘𝐾) ∈ Ring)
41		ringsrg 14401	. . . . . . . 8 ⊢ ((oppr‘𝐾) ∈ Ring → (oppr‘𝐾) ∈ SRing)
42	4, 40, 41	3syl 17	. . . . . . 7 ⊢ (𝜑 → (oppr‘𝐾) ∈ SRing)
43	21	opprring 14433	. . . . . . . 8 ⊢ (𝐿 ∈ Ring → (oppr‘𝐿) ∈ Ring)
44		ringsrg 14401	. . . . . . . 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 14503	. . . . . 6 ⊢ (𝜑 → (∥r‘(oppr‘𝐾)) = (∥r‘(oppr‘𝐿)))
47	46	breqd 4141	. . . . 5 ⊢ (𝜑 → (𝑧(∥r‘(oppr‘𝐾))(1r‘𝐿) ↔ 𝑧(∥r‘(oppr‘𝐿))(1r‘𝐿)))
48	15, 47	anbi12d 477	. . . 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 2239	. . . 4 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐾))
51		eqidd 2239	. . . 4 ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐾))
52		eqidd 2239	. . . 4 ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐾))
53		eqidd 2239	. . . 4 ⊢ (𝜑 → (oppr‘𝐾) = (oppr‘𝐾))
54		eqidd 2239	. . . 4 ⊢ (𝜑 → (∥r‘(oppr‘𝐾)) = (∥r‘(oppr‘𝐾)))
55	50, 51, 52, 53, 54, 11	isunitd 14462	. . 3 ⊢ (𝜑 → (𝑧 ∈ (Unit‘𝐾) ↔ (𝑧(∥r‘𝐾)(1r‘𝐾) ∧ 𝑧(∥r‘(oppr‘𝐾))(1r‘𝐾))))
56		eqidd 2239	. . . 4 ⊢ (𝜑 → (Unit‘𝐿) = (Unit‘𝐿))
57		eqidd 2239	. . . 4 ⊢ (𝜑 → (1r‘𝐿) = (1r‘𝐿))
58		eqidd 2239	. . . 4 ⊢ (𝜑 → (∥r‘𝐿) = (∥r‘𝐿))
59		eqidd 2239	. . . 4 ⊢ (𝜑 → (oppr‘𝐿) = (oppr‘𝐿))
60		eqidd 2239	. . . 4 ⊢ (𝜑 → (∥r‘(oppr‘𝐿)) = (∥r‘(oppr‘𝐿)))
61	56, 57, 58, 59, 60, 13	isunitd 14462	. . 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 2236	1 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209   class class class wbr 4130  ‘cfv 5377  (class class class)co 6085  Basecbs 13401  ._rcmulr 13481  1_rcur 14311  SRingcsrg 14316  Ringcrg 14349  opp_rcoppr 14421  ∥_rcdsr 14441  Unitcui 14442

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

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 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-tpos 6516  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9307  df-2 9365  df-3 9366  df-ndx 13404  df-slot 13405  df-base 13407  df-sets 13408  df-plusg 13493  df-mulr 13494  df-0g 13661  df-mgm 13725  df-sgrp 13766  df-mnd 13779  df-grp 13857  df-minusg 13858  df-cmn 14138  df-abl 14139  df-mgp 14267  df-ur 14312  df-srg 14317  df-ring 14351  df-oppr 14422  df-dvdsr 14444  df-unit 14445

This theorem is used by:  invrpropdg  14505  aprprop  14650