Theorem xpsring1d 20330

Description: The multiplicative identity element of a binary product of rings. (Contributed by AV, 16-Mar-2025.)

Hypotheses

Ref	Expression
xpsringd.y	⊢ 𝑌 = (𝑆 ×s 𝑅)
xpsringd.s	⊢ (𝜑 → 𝑆 ∈ Ring)
xpsringd.r	⊢ (𝜑 → 𝑅 ∈ Ring)

Assertion

Ref	Expression
xpsring1d	⊢ (𝜑 → (1r‘𝑌) = ⟨(1r‘𝑆), (1r‘𝑅)⟩)

Proof of Theorem xpsring1d

Dummy variables 𝑥 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . 4 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌)
2		eqid 2737	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
3	1, 2	mgpbas 20142	. . 3 ⊢ (Base‘𝑌) = (Base‘(mulGrp‘𝑌))
4		eqid 2737	. . . 4 ⊢ (1r‘𝑌) = (1r‘𝑌)
5	1, 4	ringidval 20180	. . 3 ⊢ (1r‘𝑌) = (0g‘(mulGrp‘𝑌))
6		eqid 2737	. . . 4 ⊢ (.r‘𝑌) = (.r‘𝑌)
7	1, 6	mgpplusg 20141	. . 3 ⊢ (.r‘𝑌) = (+g‘(mulGrp‘𝑌))
8		xpsringd.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Ring)
9		eqid 2737	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
10		eqid 2737	. . . . . . 7 ⊢ (1r‘𝑆) = (1r‘𝑆)
11	9, 10	ringidcl 20262	. . . . . 6 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) ∈ (Base‘𝑆))
12	8, 11	syl 17	. . . . 5 ⊢ (𝜑 → (1r‘𝑆) ∈ (Base‘𝑆))
13		xpsringd.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring)
14		eqid 2737	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
15		eqid 2737	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
16	14, 15	ringidcl 20262	. . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
17	13, 16	syl 17	. . . . 5 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
18	12, 17	opelxpd 5724	. . . 4 ⊢ (𝜑 → ⟨(1r‘𝑆), (1r‘𝑅)⟩ ∈ ((Base‘𝑆) × (Base‘𝑅)))
19		xpsringd.y	. . . . 5 ⊢ 𝑌 = (𝑆 ×s 𝑅)
20	19, 9, 14, 8, 13	xpsbas 17617	. . . 4 ⊢ (𝜑 → ((Base‘𝑆) × (Base‘𝑅)) = (Base‘𝑌))
21	18, 20	eleqtrd 2843	. . 3 ⊢ (𝜑 → ⟨(1r‘𝑆), (1r‘𝑅)⟩ ∈ (Base‘𝑌))
22	20	eleq2d 2827	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑆) × (Base‘𝑅)) ↔ 𝑥 ∈ (Base‘𝑌)))
23		elxp2 5709	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝑆) × (Base‘𝑅)) ↔ ∃𝑎 ∈ (Base‘𝑆)∃𝑏 ∈ (Base‘𝑅)𝑥 = ⟨𝑎, 𝑏⟩)
24	8	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑆 ∈ Ring)
25	13	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring)
26	12	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (1r‘𝑆) ∈ (Base‘𝑆))
27	17	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (1r‘𝑅) ∈ (Base‘𝑅))
28		simprl 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑎 ∈ (Base‘𝑆))
29		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ (Base‘𝑅))
30		eqid 2737	. . . . . . . . . . 11 ⊢ (.r‘𝑆) = (.r‘𝑆)
31	9, 30, 24, 26, 28	ringcld 20257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r‘𝑆)(.r‘𝑆)𝑎) ∈ (Base‘𝑆))
32		eqid 2737	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
33	14, 32, 25, 27, 29	ringcld 20257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r‘𝑅)(.r‘𝑅)𝑏) ∈ (Base‘𝑅))
34	19, 9, 14, 24, 25, 26, 27, 28, 29, 31, 33, 30, 32, 6	xpsmul 17620	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)⟨𝑎, 𝑏⟩) = ⟨((1r‘𝑆)(.r‘𝑆)𝑎), ((1r‘𝑅)(.r‘𝑅)𝑏)⟩)
35		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑎 ∈ (Base‘𝑆))
36	9, 30, 10	ringlidm 20266	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆)) → ((1r‘𝑆)(.r‘𝑆)𝑎) = 𝑎)
37	8, 35, 36	syl2an 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r‘𝑆)(.r‘𝑆)𝑎) = 𝑎)
38		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑏 ∈ (Base‘𝑅))
39	14, 32, 15	ringlidm 20266	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑏 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑏) = 𝑏)
40	13, 38, 39	syl2an 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r‘𝑅)(.r‘𝑅)𝑏) = 𝑏)
41	37, 40	opeq12d 4881	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ⟨((1r‘𝑆)(.r‘𝑆)𝑎), ((1r‘𝑅)(.r‘𝑅)𝑏)⟩ = ⟨𝑎, 𝑏⟩)
42	34, 41	eqtrd 2777	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)⟨𝑎, 𝑏⟩) = ⟨𝑎, 𝑏⟩)
43		oveq2 7439	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)𝑥) = (⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)⟨𝑎, 𝑏⟩))
44		id 22	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → 𝑥 = ⟨𝑎, 𝑏⟩)
45	43, 44	eqeq12d 2753	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)𝑥) = 𝑥 ↔ (⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)⟨𝑎, 𝑏⟩) = ⟨𝑎, 𝑏⟩))
46	42, 45	syl5ibrcom 247	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑥 = ⟨𝑎, 𝑏⟩ → (⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)𝑥) = 𝑥))
47	46	rexlimdvva 3213	. . . . . 6 ⊢ (𝜑 → (∃𝑎 ∈ (Base‘𝑆)∃𝑏 ∈ (Base‘𝑅)𝑥 = ⟨𝑎, 𝑏⟩ → (⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)𝑥) = 𝑥))
48	23, 47	biimtrid 242	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑆) × (Base‘𝑅)) → (⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)𝑥) = 𝑥))
49	22, 48	sylbird 260	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑌) → (⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)𝑥) = 𝑥))
50	49	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → (⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)𝑥) = 𝑥)
51	9, 30, 24, 28, 26	ringcld 20257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(.r‘𝑆)(1r‘𝑆)) ∈ (Base‘𝑆))
52	14, 32, 25, 29, 27	ringcld 20257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑏(.r‘𝑅)(1r‘𝑅)) ∈ (Base‘𝑅))
53	19, 9, 14, 24, 25, 28, 29, 26, 27, 51, 52, 30, 32, 6	xpsmul 17620	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨𝑎, 𝑏⟩(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = ⟨(𝑎(.r‘𝑆)(1r‘𝑆)), (𝑏(.r‘𝑅)(1r‘𝑅))⟩)
54	9, 30, 10	ringridm 20267	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑎(.r‘𝑆)(1r‘𝑆)) = 𝑎)
55	8, 35, 54	syl2an 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(.r‘𝑆)(1r‘𝑆)) = 𝑎)
56	14, 32, 15	ringridm 20267	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑏(.r‘𝑅)(1r‘𝑅)) = 𝑏)
57	13, 38, 56	syl2an 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑏(.r‘𝑅)(1r‘𝑅)) = 𝑏)
58	55, 57	opeq12d 4881	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ⟨(𝑎(.r‘𝑆)(1r‘𝑆)), (𝑏(.r‘𝑅)(1r‘𝑅))⟩ = ⟨𝑎, 𝑏⟩)
59	53, 58	eqtrd 2777	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨𝑎, 𝑏⟩(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = ⟨𝑎, 𝑏⟩)
60		oveq1 7438	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = (⟨𝑎, 𝑏⟩(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩))
61	60, 44	eqeq12d 2753	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝑥(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = 𝑥 ↔ (⟨𝑎, 𝑏⟩(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = ⟨𝑎, 𝑏⟩))
62	59, 61	syl5ibrcom 247	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = 𝑥))
63	62	rexlimdvva 3213	. . . . . 6 ⊢ (𝜑 → (∃𝑎 ∈ (Base‘𝑆)∃𝑏 ∈ (Base‘𝑅)𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = 𝑥))
64	23, 63	biimtrid 242	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑆) × (Base‘𝑅)) → (𝑥(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = 𝑥))
65	22, 64	sylbird 260	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑌) → (𝑥(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = 𝑥))
66	65	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → (𝑥(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = 𝑥)
67	3, 5, 7, 21, 50, 66	ismgmid2 18681	. 2 ⊢ (𝜑 → ⟨(1r‘𝑆), (1r‘𝑅)⟩ = (1r‘𝑌))
68	67	eqcomd 2743	1 ⊢ (𝜑 → (1r‘𝑌) = ⟨(1r‘𝑆), (1r‘𝑅)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  ⟨cop 4632   × cxp 5683  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  ._rcmulr 17298   ×_s cxps 17551  mulGrpcmgp 20137  1_rcur 20178  Ringcrg 20230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-prds 17492  df-imas 17553  df-xps 17555  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mgp 20138  df-ur 20179  df-ring 20232

This theorem is referenced by:  rngqipring1  21326  pzriprng1  21509