Theorem xpsring1d 20298

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 20110	. . 3 ⊢ (Base‘𝑌) = (Base‘(mulGrp‘𝑌))
4		eqid 2736	. . . 4 ⊢ (1r‘𝑌) = (1r‘𝑌)
5	1, 4	ringidval 20148	. . 3 ⊢ (1r‘𝑌) = (0g‘(mulGrp‘𝑌))
6		eqid 2736	. . . 4 ⊢ (.r‘𝑌) = (.r‘𝑌)
7	1, 6	mgpplusg 20109	. . 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 20230	. . . . . 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 20230	. . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
17	13, 16	syl 17	. . . . 5 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
18	12, 17	opelxpd 5698	. . . 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 17591	. . . 4 ⊢ (𝜑 → ((Base‘𝑆) × (Base‘𝑅)) = (Base‘𝑌))
21	18, 20	eleqtrd 2837	. . 3 ⊢ (𝜑 → ⟨(1r‘𝑆), (1r‘𝑅)⟩ ∈ (Base‘𝑌))
22	20	eleq2d 2821	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑆) × (Base‘𝑅)) ↔ 𝑥 ∈ (Base‘𝑌)))
23		elxp2 5683	. . . . . 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 20225	. . . . . . . . . 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 20225	. . . . . . . . . 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 17594	. . . . . . . . 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 20234	. . . . . . . . . . 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 20234	. . . . . . . . . . 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 4862	. . . . . . . . 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 7418	. . . . . . . . 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 3202	. . . . . 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 20225	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(.r‘𝑆)(1r‘𝑆)) ∈ (Base‘𝑆))
52	14, 32, 25, 29, 27	ringcld 20225	. . . . . . . . . 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 17594	. . . . . . . . 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 20235	. . . . . . . . . . 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 20235	. . . . . . . . . . 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 4862	. . . . . . . . 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 7417	. . . . . . . . 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 3202	. . . . . 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 18651	. 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 1540   ∈ wcel 2109  ∃wrex 3061  ⟨cop 4612   × cxp 5657  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  ._rcmulr 17277   ×_s cxps 17525  mulGrpcmgp 20105  1_rcur 20146  Ringcrg 20198

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-er 8724  df-map 8847  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9459  df-inf 9460  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12858  df-fz 13530  df-struct 17171  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-plusg 17289  df-mulr 17290  df-sca 17292  df-vsca 17293  df-ip 17294  df-tset 17295  df-ple 17296  df-ds 17298  df-hom 17300  df-cco 17301  df-0g 17460  df-prds 17466  df-imas 17527  df-xps 17529  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-mgp 20106  df-ur 20147  df-ring 20200

This theorem is referenced by:  rngqipring1  21282  pzriprng1  21464