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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 (𝜑𝑆 ∈ Ring)
xpsringd.r (𝜑𝑅 ∈ Ring)
Assertion
Ref Expression
xpsring1d (𝜑 → (1r𝑌) = ⟨(1r𝑆), (1r𝑅)⟩)

Proof of Theorem xpsring1d
Dummy variables 𝑥 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . 4 (mulGrp‘𝑌) = (mulGrp‘𝑌)
2 eqid 2737 . . . 4 (Base‘𝑌) = (Base‘𝑌)
31, 2mgpbas 20142 . . 3 (Base‘𝑌) = (Base‘(mulGrp‘𝑌))
4 eqid 2737 . . . 4 (1r𝑌) = (1r𝑌)
51, 4ringidval 20180 . . 3 (1r𝑌) = (0g‘(mulGrp‘𝑌))
6 eqid 2737 . . . 4 (.r𝑌) = (.r𝑌)
71, 6mgpplusg 20141 . . 3 (.r𝑌) = (+g‘(mulGrp‘𝑌))
8 xpsringd.s . . . . . 6 (𝜑𝑆 ∈ Ring)
9 eqid 2737 . . . . . . 7 (Base‘𝑆) = (Base‘𝑆)
10 eqid 2737 . . . . . . 7 (1r𝑆) = (1r𝑆)
119, 10ringidcl 20262 . . . . . 6 (𝑆 ∈ Ring → (1r𝑆) ∈ (Base‘𝑆))
128, 11syl 17 . . . . 5 (𝜑 → (1r𝑆) ∈ (Base‘𝑆))
13 xpsringd.r . . . . . 6 (𝜑𝑅 ∈ Ring)
14 eqid 2737 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
15 eqid 2737 . . . . . . 7 (1r𝑅) = (1r𝑅)
1614, 15ringidcl 20262 . . . . . 6 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
1713, 16syl 17 . . . . 5 (𝜑 → (1r𝑅) ∈ (Base‘𝑅))
1812, 17opelxpd 5724 . . . 4 (𝜑 → ⟨(1r𝑆), (1r𝑅)⟩ ∈ ((Base‘𝑆) × (Base‘𝑅)))
19 xpsringd.y . . . . 5 𝑌 = (𝑆 ×s 𝑅)
2019, 9, 14, 8, 13xpsbas 17617 . . . 4 (𝜑 → ((Base‘𝑆) × (Base‘𝑅)) = (Base‘𝑌))
2118, 20eleqtrd 2843 . . 3 (𝜑 → ⟨(1r𝑆), (1r𝑅)⟩ ∈ (Base‘𝑌))
2220eleq2d 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⟩)
248adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑆 ∈ Ring)
2513adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring)
2612adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (1r𝑆) ∈ (Base‘𝑆))
2717adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (1r𝑅) ∈ (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 (.r𝑆) = (.r𝑆)
319, 30, 24, 26, 28ringcld 20257 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r𝑆)(.r𝑆)𝑎) ∈ (Base‘𝑆))
32 eqid 2737 . . . . . . . . . . 11 (.r𝑅) = (.r𝑅)
3314, 32, 25, 27, 29ringcld 20257 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r𝑅)(.r𝑅)𝑏) ∈ (Base‘𝑅))
3419, 9, 14, 24, 25, 26, 27, 28, 29, 31, 33, 30, 32, 6xpsmul 17620 . . . . . . . . 9 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨(1r𝑆), (1r𝑅)⟩(.r𝑌)⟨𝑎, 𝑏⟩) = ⟨((1r𝑆)(.r𝑆)𝑎), ((1r𝑅)(.r𝑅)𝑏)⟩)
35 simpl 482 . . . . . . . . . . 11 ((𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑎 ∈ (Base‘𝑆))
369, 30, 10ringlidm 20266 . . . . . . . . . . 11 ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆)) → ((1r𝑆)(.r𝑆)𝑎) = 𝑎)
378, 35, 36syl2an 596 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r𝑆)(.r𝑆)𝑎) = 𝑎)
38 simpr 484 . . . . . . . . . . 11 ((𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅)) → 𝑏 ∈ (Base‘𝑅))
3914, 32, 15ringlidm 20266 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑏 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑏) = 𝑏)
4013, 38, 39syl2an 596 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r𝑅)(.r𝑅)𝑏) = 𝑏)
4137, 40opeq12d 4881 . . . . . . . . 9 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ⟨((1r𝑆)(.r𝑆)𝑎), ((1r𝑅)(.r𝑅)𝑏)⟩ = ⟨𝑎, 𝑏⟩)
4234, 41eqtrd 2777 . . . . . . . 8 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨(1r𝑆), (1r𝑅)⟩(.r𝑌)⟨𝑎, 𝑏⟩) = ⟨𝑎, 𝑏⟩)
43 oveq2 7439 . . . . . . . . 9 (𝑥 = ⟨𝑎, 𝑏⟩ → (⟨(1r𝑆), (1r𝑅)⟩(.r𝑌)𝑥) = (⟨(1r𝑆), (1r𝑅)⟩(.r𝑌)⟨𝑎, 𝑏⟩))
44 id 22 . . . . . . . . 9 (𝑥 = ⟨𝑎, 𝑏⟩ → 𝑥 = ⟨𝑎, 𝑏⟩)
4543, 44eqeq12d 2753 . . . . . . . 8 (𝑥 = ⟨𝑎, 𝑏⟩ → ((⟨(1r𝑆), (1r𝑅)⟩(.r𝑌)𝑥) = 𝑥 ↔ (⟨(1r𝑆), (1r𝑅)⟩(.r𝑌)⟨𝑎, 𝑏⟩) = ⟨𝑎, 𝑏⟩))
4642, 45syl5ibrcom 247 . . . . . . 7 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑥 = ⟨𝑎, 𝑏⟩ → (⟨(1r𝑆), (1r𝑅)⟩(.r𝑌)𝑥) = 𝑥))
4746rexlimdvva 3213 . . . . . 6 (𝜑 → (∃𝑎 ∈ (Base‘𝑆)∃𝑏 ∈ (Base‘𝑅)𝑥 = ⟨𝑎, 𝑏⟩ → (⟨(1r𝑆), (1r𝑅)⟩(.r𝑌)𝑥) = 𝑥))
4823, 47biimtrid 242 . . . . 5 (𝜑 → (𝑥 ∈ ((Base‘𝑆) × (Base‘𝑅)) → (⟨(1r𝑆), (1r𝑅)⟩(.r𝑌)𝑥) = 𝑥))
4922, 48sylbird 260 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝑌) → (⟨(1r𝑆), (1r𝑅)⟩(.r𝑌)𝑥) = 𝑥))
5049imp 406 . . 3 ((𝜑𝑥 ∈ (Base‘𝑌)) → (⟨(1r𝑆), (1r𝑅)⟩(.r𝑌)𝑥) = 𝑥)
519, 30, 24, 28, 26ringcld 20257 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(.r𝑆)(1r𝑆)) ∈ (Base‘𝑆))
5214, 32, 25, 29, 27ringcld 20257 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑏(.r𝑅)(1r𝑅)) ∈ (Base‘𝑅))
5319, 9, 14, 24, 25, 28, 29, 26, 27, 51, 52, 30, 32, 6xpsmul 17620 . . . . . . . . 9 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨𝑎, 𝑏⟩(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = ⟨(𝑎(.r𝑆)(1r𝑆)), (𝑏(.r𝑅)(1r𝑅))⟩)
549, 30, 10ringridm 20267 . . . . . . . . . . 11 ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑎(.r𝑆)(1r𝑆)) = 𝑎)
558, 35, 54syl2an 596 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(.r𝑆)(1r𝑆)) = 𝑎)
5614, 32, 15ringridm 20267 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑏(.r𝑅)(1r𝑅)) = 𝑏)
5713, 38, 56syl2an 596 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑏(.r𝑅)(1r𝑅)) = 𝑏)
5855, 57opeq12d 4881 . . . . . . . . 9 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ⟨(𝑎(.r𝑆)(1r𝑆)), (𝑏(.r𝑅)(1r𝑅))⟩ = ⟨𝑎, 𝑏⟩)
5953, 58eqtrd 2777 . . . . . . . 8 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨𝑎, 𝑏⟩(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = ⟨𝑎, 𝑏⟩)
60 oveq1 7438 . . . . . . . . 9 (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = (⟨𝑎, 𝑏⟩(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩))
6160, 44eqeq12d 2753 . . . . . . . 8 (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝑥(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = 𝑥 ↔ (⟨𝑎, 𝑏⟩(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = ⟨𝑎, 𝑏⟩))
6259, 61syl5ibrcom 247 . . . . . . 7 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = 𝑥))
6362rexlimdvva 3213 . . . . . 6 (𝜑 → (∃𝑎 ∈ (Base‘𝑆)∃𝑏 ∈ (Base‘𝑅)𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = 𝑥))
6423, 63biimtrid 242 . . . . 5 (𝜑 → (𝑥 ∈ ((Base‘𝑆) × (Base‘𝑅)) → (𝑥(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = 𝑥))
6522, 64sylbird 260 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝑌) → (𝑥(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = 𝑥))
6665imp 406 . . 3 ((𝜑𝑥 ∈ (Base‘𝑌)) → (𝑥(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = 𝑥)
673, 5, 7, 21, 50, 66ismgmid2 18681 . 2 (𝜑 → ⟨(1r𝑆), (1r𝑅)⟩ = (1r𝑌))
6867eqcomd 2743 1 (𝜑 → (1r𝑌) = ⟨(1r𝑆), (1r𝑅)⟩)
Colors of variables: wff setvar class
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  1rcur 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
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