Theorem xpsring1d 20269

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 2736	. . . 4 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌)
2		eqid 2736	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
3	1, 2	mgpbas 20080	. . 3 ⊢ (Base‘𝑌) = (Base‘(mulGrp‘𝑌))
4		eqid 2736	. . . 4 ⊢ (1r‘𝑌) = (1r‘𝑌)
5	1, 4	ringidval 20118	. . 3 ⊢ (1r‘𝑌) = (0g‘(mulGrp‘𝑌))
6		eqid 2736	. . . 4 ⊢ (.r‘𝑌) = (.r‘𝑌)
7	1, 6	mgpplusg 20079	. . 3 ⊢ (.r‘𝑌) = (+g‘(mulGrp‘𝑌))
8		xpsringd.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Ring)
9		eqid 2736	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
10		eqid 2736	. . . . . . 7 ⊢ (1r‘𝑆) = (1r‘𝑆)
11	9, 10	ringidcl 20200	. . . . . 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 2736	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
15		eqid 2736	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
16	14, 15	ringidcl 20200	. . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
17	13, 16	syl 17	. . . . 5 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
18	12, 17	opelxpd 5663	. . . 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 17493	. . . 4 ⊢ (𝜑 → ((Base‘𝑆) × (Base‘𝑅)) = (Base‘𝑌))
21	18, 20	eleqtrd 2838	. . 3 ⊢ (𝜑 → ⟨(1r‘𝑆), (1r‘𝑅)⟩ ∈ (Base‘𝑌))
22	20	eleq2d 2822	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑆) × (Base‘𝑅)) ↔ 𝑥 ∈ (Base‘𝑌)))
23		elxp2 5648	. . . . . 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 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑎 ∈ (Base‘𝑆))
29		simprr 772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ (Base‘𝑅))
30		eqid 2736	. . . . . . . . . . 11 ⊢ (.r‘𝑆) = (.r‘𝑆)
31	9, 30, 24, 26, 28	ringcld 20195	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r‘𝑆)(.r‘𝑆)𝑎) ∈ (Base‘𝑆))
32		eqid 2736	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
33	14, 32, 25, 27, 29	ringcld 20195	. . . . . . . . . 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 17496	. . . . . . . . 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 20204	. . . . . . . . . . 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 20204	. . . . . . . . . . 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 4837	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ⟨((1r‘𝑆)(.r‘𝑆)𝑎), ((1r‘𝑅)(.r‘𝑅)𝑏)⟩ = ⟨𝑎, 𝑏⟩)
42	34, 41	eqtrd 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)⟨𝑎, 𝑏⟩) = ⟨𝑎, 𝑏⟩)
43		oveq2 7366	. . . . . . . . 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 2752	. . . . . . . 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 3193	. . . . . 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 20195	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(.r‘𝑆)(1r‘𝑆)) ∈ (Base‘𝑆))
52	14, 32, 25, 29, 27	ringcld 20195	. . . . . . . . . 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 17496	. . . . . . . . 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 20205	. . . . . . . . . . 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 20205	. . . . . . . . . . 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 4837	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ⟨(𝑎(.r‘𝑆)(1r‘𝑆)), (𝑏(.r‘𝑅)(1r‘𝑅))⟩ = ⟨𝑎, 𝑏⟩)
59	53, 58	eqtrd 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨𝑎, 𝑏⟩(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = ⟨𝑎, 𝑏⟩)
60		oveq1 7365	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = (⟨𝑎, 𝑏⟩(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩))
61	60, 44	eqeq12d 2752	. . . . . . . 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 3193	. . . . . 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 18593	. 2 ⊢ (𝜑 → ⟨(1r‘𝑆), (1r‘𝑅)⟩ = (1r‘𝑌))
68	67	eqcomd 2742	1 ⊢ (𝜑 → (1r‘𝑌) = ⟨(1r‘𝑆), (1r‘𝑅)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3060  ⟨cop 4586   × cxp 5622  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  ._rcmulr 17178   ×_s cxps 17427  mulGrpcmgp 20075  1_rcur 20116  Ringcrg 20168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-plusg 17190  df-mulr 17191  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ds 17199  df-hom 17201  df-cco 17202  df-0g 17361  df-prds 17367  df-imas 17429  df-xps 17431  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mgp 20076  df-ur 20117  df-ring 20170

This theorem is referenced by:  rngqipring1  21271  pzriprng1  21453