Theorem xpsring1d 20596

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 2763	. . . 4 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌)
2		eqid 2763	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
3	1, 2	mgpbas 20398	. . 3 ⊢ (Base‘𝑌) = (Base‘(mulGrp‘𝑌))
4		eqid 2763	. . . 4 ⊢ (1r‘𝑌) = (1r‘𝑌)
5	1, 4	ringidval 20442	. . 3 ⊢ (1r‘𝑌) = (0g‘(mulGrp‘𝑌))
6		eqid 2763	. . . 4 ⊢ (.r‘𝑌) = (.r‘𝑌)
7	1, 6	mgpplusg 20397	. . 3 ⊢ (.r‘𝑌) = (+g‘(mulGrp‘𝑌))
8		xpsringd.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Ring)
9		eqid 2763	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
10		eqid 2763	. . . . . . 7 ⊢ (1r‘𝑆) = (1r‘𝑆)
11	9, 10	ringidcl 20527	. . . . . 6 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) ∈ (Base‘𝑆))
12	8, 11	syl 18	. . . . 5 ⊢ (𝜑 → (1r‘𝑆) ∈ (Base‘𝑆))
13		xpsringd.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring)
14		eqid 2763	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
15		eqid 2763	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
16	14, 15	ringidcl 20527	. . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
17	13, 16	syl 18	. . . . 5 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
18	12, 17	opelxpd 5695	. . . 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 17776	. . . 4 ⊢ (𝜑 → ((Base‘𝑆) × (Base‘𝑅)) = (Base‘𝑌))
21	18, 20	eleqtrd 2866	. . 3 ⊢ (𝜑 → ⟨(1r‘𝑆), (1r‘𝑅)⟩ ∈ (Base‘𝑌))
22	20	eleq2d 2850	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑆) × (Base‘𝑅)) ↔ 𝑥 ∈ (Base‘𝑌)))
23		elxp2 5680	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝑆) × (Base‘𝑅)) ↔ ∃𝑎 ∈ (Base‘𝑆)∃𝑏 ∈ (Base‘𝑅)𝑥 = ⟨𝑎, 𝑏⟩)
24	8	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑆 ∈ Ring)
25	13	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring)
26	12	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (1r‘𝑆) ∈ (Base‘𝑆))
27	17	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (1r‘𝑅) ∈ (Base‘𝑅))
28		simprl 783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑎 ∈ (Base‘𝑆))
29		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ (Base‘𝑅))
30		eqid 2763	. . . . . . . . . . 11 ⊢ (.r‘𝑆) = (.r‘𝑆)
31	9, 30, 24, 26, 28	ringcld 20517	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r‘𝑆)(.r‘𝑆)𝑎) ∈ (Base‘𝑆))
32		eqid 2763	. . . . . . . . . . 11 ⊢ (.r‘𝑅) = (.r‘𝑅)
33	14, 32, 25, 27, 29	ringcld 20517	. . . . . . . . . 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 17779	. . . . . . . . 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 488	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑎 ∈ (Base‘𝑆))
36	9, 30, 10	ringlidm 20531	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆)) → ((1r‘𝑆)(.r‘𝑆)𝑎) = 𝑎)
37	8, 35, 36	syl2an 608	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r‘𝑆)(.r‘𝑆)𝑎) = 𝑎)
38		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑏 ∈ (Base‘𝑅))
39	14, 32, 15	ringlidm 20531	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑏 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑏) = 𝑏)
40	13, 38, 39	syl2an 608	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r‘𝑅)(.r‘𝑅)𝑏) = 𝑏)
41	37, 40	opeq12d 4851	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ⟨((1r‘𝑆)(.r‘𝑆)𝑎), ((1r‘𝑅)(.r‘𝑅)𝑏)⟩ = ⟨𝑎, 𝑏⟩)
42	34, 41	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)⟨𝑎, 𝑏⟩) = ⟨𝑎, 𝑏⟩)
43		oveq2 7454	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)𝑥) = (⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)⟨𝑎, 𝑏⟩))
44		id 23	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → 𝑥 = ⟨𝑎, 𝑏⟩)
45	43, 44	eqeq12d 2779	. . . . . . . 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 250	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑥 = ⟨𝑎, 𝑏⟩ → (⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)𝑥) = 𝑥))
47	46	rexlimdvva 3223	. . . . . 6 ⊢ (𝜑 → (∃𝑎 ∈ (Base‘𝑆)∃𝑏 ∈ (Base‘𝑅)𝑥 = ⟨𝑎, 𝑏⟩ → (⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)𝑥) = 𝑥))
48	23, 47	biimtrid 245	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑆) × (Base‘𝑅)) → (⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)𝑥) = 𝑥))
49	22, 48	sylbird 263	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑌) → (⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)𝑥) = 𝑥))
50	49	imp 412	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → (⟨(1r‘𝑆), (1r‘𝑅)⟩(.r‘𝑌)𝑥) = 𝑥)
51	9, 30, 24, 28, 26	ringcld 20517	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(.r‘𝑆)(1r‘𝑆)) ∈ (Base‘𝑆))
52	14, 32, 25, 29, 27	ringcld 20517	. . . . . . . . . 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 17779	. . . . . . . . 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 20532	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑎(.r‘𝑆)(1r‘𝑆)) = 𝑎)
55	8, 35, 54	syl2an 608	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(.r‘𝑆)(1r‘𝑆)) = 𝑎)
56	14, 32, 15	ringridm 20532	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑏(.r‘𝑅)(1r‘𝑅)) = 𝑏)
57	13, 38, 56	syl2an 608	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑏(.r‘𝑅)(1r‘𝑅)) = 𝑏)
58	55, 57	opeq12d 4851	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ⟨(𝑎(.r‘𝑆)(1r‘𝑆)), (𝑏(.r‘𝑅)(1r‘𝑅))⟩ = ⟨𝑎, 𝑏⟩)
59	53, 58	eqtrd 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨𝑎, 𝑏⟩(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = ⟨𝑎, 𝑏⟩)
60		oveq1 7453	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = (⟨𝑎, 𝑏⟩(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩))
61	60, 44	eqeq12d 2779	. . . . . . . 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 250	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = 𝑥))
63	62	rexlimdvva 3223	. . . . . 6 ⊢ (𝜑 → (∃𝑎 ∈ (Base‘𝑆)∃𝑏 ∈ (Base‘𝑅)𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = 𝑥))
64	23, 63	biimtrid 245	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((Base‘𝑆) × (Base‘𝑅)) → (𝑥(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = 𝑥))
65	22, 64	sylbird 263	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑌) → (𝑥(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = 𝑥))
66	65	imp 412	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → (𝑥(.r‘𝑌)⟨(1r‘𝑆), (1r‘𝑅)⟩) = 𝑥)
67	3, 5, 7, 21, 50, 66	ismgmid2 18881	. 2 ⊢ (𝜑 → ⟨(1r‘𝑆), (1r‘𝑅)⟩ = (1r‘𝑌))
68	67	eqcomd 2769	1 ⊢ (𝜑 → (1r‘𝑌) = ⟨(1r‘𝑆), (1r‘𝑅)⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  ⟨cop 4838   × cxp 5654  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  ._rcmulr 17461   ×_s cxps 17710  mulGrpcmgp 20393  1_rcur 20440  Ringcrg 20492

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-2o 8508  df-er 8748  df-map 8880  df-ixp 8957  df-en 9005  df-dom 9006  df-sdom 9007  df-fin 9008  df-sup 9464  df-inf 9465  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-nn 12366  df-2 12435  df-3 12436  df-4 12437  df-5 12438  df-6 12439  df-7 12440  df-8 12441  df-9 12442  df-n0 12637  df-z 12724  df-dec 12845  df-uz 12996  df-fz 13670  df-struct 17357  df-sets 17374  df-slot 17392  df-ndx 17404  df-base 17420  df-plusg 17473  df-mulr 17474  df-sca 17476  df-vsca 17477  df-ip 17478  df-tset 17479  df-ple 17480  df-ds 17482  df-hom 17484  df-cco 17485  df-0g 17644  df-prds 17650  df-imas 17712  df-xps 17714  df-mgm 18848  df-sgrp 18940  df-mnd 18956  df-mgp 20394  df-ur 20441  df-ring 20494

This theorem is used by:  rngqipring1  21645  pzriprng1  21837