Theorem xpsring1d 20253

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 2733	. . . 4 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌)
2		eqid 2733	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
3	1, 2	mgpbas 20065	. . 3 ⊢ (Base‘𝑌) = (Base‘(mulGrp‘𝑌))
4		eqid 2733	. . . 4 ⊢ (1r‘𝑌) = (1r‘𝑌)
5	1, 4	ringidval 20103	. . 3 ⊢ (1r‘𝑌) = (0g‘(mulGrp‘𝑌))
6		eqid 2733	. . . 4 ⊢ (.r‘𝑌) = (.r‘𝑌)
7	1, 6	mgpplusg 20064	. . 3 ⊢ (.r‘𝑌) = (+g‘(mulGrp‘𝑌))
8		xpsringd.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Ring)
9		eqid 2733	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
10		eqid 2733	. . . . . . 7 ⊢ (1r‘𝑆) = (1r‘𝑆)
11	9, 10	ringidcl 20185	. . . . . 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 2733	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
15		eqid 2733	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
16	14, 15	ringidcl 20185	. . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
17	13, 16	syl 17	. . . . 5 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
18	12, 17	opelxpd 5658	. . . 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 17478	. . . 4 ⊢ (𝜑 → ((Base‘𝑆) × (Base‘𝑅)) = (Base‘𝑌))
21	18, 20	eleqtrd 2835	. . 3 ⊢ (𝜑 → ⟨(1r‘𝑆), (1r‘𝑅)⟩ ∈ (Base‘𝑌))
22	20	eleq2d 2819	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑆) × (Base‘𝑅)) ↔ 𝑥 ∈ (Base‘𝑌)))
23		elxp2 5643	. . . . . 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 2733	. . . . . . . . . . 11 ⊢ (.r‘𝑆) = (.r‘𝑆)
31	9, 30, 24, 26, 28	ringcld 20180	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r‘𝑆)(.r‘𝑆)𝑎) ∈ (Base‘𝑆))
32		eqid 2733	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
33	14, 32, 25, 27, 29	ringcld 20180	. . . . . . . . . 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 17481	. . . . . . . . 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 20189	. . . . . . . . . . 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 20189	. . . . . . . . . . 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 4832	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ⟨((1r‘𝑆)(.r‘𝑆)𝑎), ((1r‘𝑅)(.r‘𝑅)𝑏)⟩ = ⟨𝑎, 𝑏⟩)
42	34, 41	eqtrd 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)⟨𝑎, 𝑏⟩) = ⟨𝑎, 𝑏⟩)
43		oveq2 7360	. . . . . . . . 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 2749	. . . . . . . 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 3190	. . . . . 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 20180	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(.r‘𝑆)(1r‘𝑆)) ∈ (Base‘𝑆))
52	14, 32, 25, 29, 27	ringcld 20180	. . . . . . . . . 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 17481	. . . . . . . . 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 20190	. . . . . . . . . . 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 20190	. . . . . . . . . . 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 4832	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ⟨(𝑎(.r‘𝑆)(1r‘𝑆)), (𝑏(.r‘𝑅)(1r‘𝑅))⟩ = ⟨𝑎, 𝑏⟩)
59	53, 58	eqtrd 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨𝑎, 𝑏⟩(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = ⟨𝑎, 𝑏⟩)
60		oveq1 7359	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = (⟨𝑎, 𝑏⟩(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩))
61	60, 44	eqeq12d 2749	. . . . . . . 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 3190	. . . . . 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 18578	. 2 ⊢ (𝜑 → ⟨(1r‘𝑆), (1r‘𝑅)⟩ = (1r‘𝑌))
68	67	eqcomd 2739	1 ⊢ (𝜑 → (1r‘𝑌) = ⟨(1r‘𝑆), (1r‘𝑅)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3057  ⟨cop 4581   × cxp 5617  ‘cfv 6486  (class class class)co 7352  Basecbs 17122  ._rcmulr 17164   ×_s cxps 17412  mulGrpcmgp 20060  1_rcur 20101  Ringcrg 20153

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 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-er 8628  df-map 8758  df-ixp 8828  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-sup 9333  df-inf 9334  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-2 12195  df-3 12196  df-4 12197  df-5 12198  df-6 12199  df-7 12200  df-8 12201  df-9 12202  df-n0 12389  df-z 12476  df-dec 12595  df-uz 12739  df-fz 13410  df-struct 17060  df-sets 17077  df-slot 17095  df-ndx 17107  df-base 17123  df-plusg 17176  df-mulr 17177  df-sca 17179  df-vsca 17180  df-ip 17181  df-tset 17182  df-ple 17183  df-ds 17185  df-hom 17187  df-cco 17188  df-0g 17347  df-prds 17353  df-imas 17414  df-xps 17416  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-mgp 20061  df-ur 20102  df-ring 20155

This theorem is referenced by:  rngqipring1  21255  pzriprng1  21437