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Theorem xpsring1d 20228
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 2724 . . . 4 (mulGrp‘𝑌) = (mulGrp‘𝑌)
2 eqid 2724 . . . 4 (Base‘𝑌) = (Base‘𝑌)
31, 2mgpbas 20041 . . 3 (Base‘𝑌) = (Base‘(mulGrp‘𝑌))
4 eqid 2724 . . . 4 (1r𝑌) = (1r𝑌)
51, 4ringidval 20084 . . 3 (1r𝑌) = (0g‘(mulGrp‘𝑌))
6 eqid 2724 . . . 4 (.r𝑌) = (.r𝑌)
71, 6mgpplusg 20039 . . 3 (.r𝑌) = (+g‘(mulGrp‘𝑌))
8 xpsringd.s . . . . . 6 (𝜑𝑆 ∈ Ring)
9 eqid 2724 . . . . . . 7 (Base‘𝑆) = (Base‘𝑆)
10 eqid 2724 . . . . . . 7 (1r𝑆) = (1r𝑆)
119, 10ringidcl 20161 . . . . . 6 (𝑆 ∈ Ring → (1r𝑆) ∈ (Base‘𝑆))
128, 11syl 17 . . . . 5 (𝜑 → (1r𝑆) ∈ (Base‘𝑆))
13 xpsringd.r . . . . . 6 (𝜑𝑅 ∈ Ring)
14 eqid 2724 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
15 eqid 2724 . . . . . . 7 (1r𝑅) = (1r𝑅)
1614, 15ringidcl 20161 . . . . . 6 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
1713, 16syl 17 . . . . 5 (𝜑 → (1r𝑅) ∈ (Base‘𝑅))
1812, 17opelxpd 5706 . . . 4 (𝜑 → ⟨(1r𝑆), (1r𝑅)⟩ ∈ ((Base‘𝑆) × (Base‘𝑅)))
19 xpsringd.y . . . . 5 𝑌 = (𝑆 ×s 𝑅)
2019, 9, 14, 8, 13xpsbas 17523 . . . 4 (𝜑 → ((Base‘𝑆) × (Base‘𝑅)) = (Base‘𝑌))
2118, 20eleqtrd 2827 . . 3 (𝜑 → ⟨(1r𝑆), (1r𝑅)⟩ ∈ (Base‘𝑌))
2220eleq2d 2811 . . . . 5 (𝜑 → (𝑥 ∈ ((Base‘𝑆) × (Base‘𝑅)) ↔ 𝑥 ∈ (Base‘𝑌)))
23 elxp2 5691 . . . . . 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 768 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑎 ∈ (Base‘𝑆))
29 simprr 770 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → 𝑏 ∈ (Base‘𝑅))
30 eqid 2724 . . . . . . . . . . 11 (.r𝑆) = (.r𝑆)
319, 30, 24, 26, 28ringcld 20158 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r𝑆)(.r𝑆)𝑎) ∈ (Base‘𝑆))
32 eqid 2724 . . . . . . . . . . 11 (.r𝑅) = (.r𝑅)
3314, 32, 25, 27, 29ringcld 20158 . . . . . . . . . 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 17526 . . . . . . . . 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 20164 . . . . . . . . . . 11 ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆)) → ((1r𝑆)(.r𝑆)𝑎) = 𝑎)
378, 35, 36syl2an 595 . . . . . . . . . 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 20164 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑏 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑏) = 𝑏)
4013, 38, 39syl2an 595 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r𝑅)(.r𝑅)𝑏) = 𝑏)
4137, 40opeq12d 4874 . . . . . . . . 9 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ⟨((1r𝑆)(.r𝑆)𝑎), ((1r𝑅)(.r𝑅)𝑏)⟩ = ⟨𝑎, 𝑏⟩)
4234, 41eqtrd 2764 . . . . . . . 8 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨(1r𝑆), (1r𝑅)⟩(.r𝑌)⟨𝑎, 𝑏⟩) = ⟨𝑎, 𝑏⟩)
43 oveq2 7410 . . . . . . . . 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 2740 . . . . . . . 8 (𝑥 = ⟨𝑎, 𝑏⟩ → ((⟨(1r𝑆), (1r𝑅)⟩(.r𝑌)𝑥) = 𝑥 ↔ (⟨(1r𝑆), (1r𝑅)⟩(.r𝑌)⟨𝑎, 𝑏⟩) = ⟨𝑎, 𝑏⟩))
4642, 45syl5ibrcom 246 . . . . . . 7 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑥 = ⟨𝑎, 𝑏⟩ → (⟨(1r𝑆), (1r𝑅)⟩(.r𝑌)𝑥) = 𝑥))
4746rexlimdvva 3203 . . . . . 6 (𝜑 → (∃𝑎 ∈ (Base‘𝑆)∃𝑏 ∈ (Base‘𝑅)𝑥 = ⟨𝑎, 𝑏⟩ → (⟨(1r𝑆), (1r𝑅)⟩(.r𝑌)𝑥) = 𝑥))
4823, 47biimtrid 241 . . . . 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 20158 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(.r𝑆)(1r𝑆)) ∈ (Base‘𝑆))
5214, 32, 25, 29, 27ringcld 20158 . . . . . . . . . 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 17526 . . . . . . . . 9 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨𝑎, 𝑏⟩(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = ⟨(𝑎(.r𝑆)(1r𝑆)), (𝑏(.r𝑅)(1r𝑅))⟩)
549, 30, 10ringridm 20165 . . . . . . . . . . 11 ((𝑆 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑆)) → (𝑎(.r𝑆)(1r𝑆)) = 𝑎)
558, 35, 54syl2an 595 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(.r𝑆)(1r𝑆)) = 𝑎)
5614, 32, 15ringridm 20165 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑏(.r𝑅)(1r𝑅)) = 𝑏)
5713, 38, 56syl2an 595 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑏(.r𝑅)(1r𝑅)) = 𝑏)
5855, 57opeq12d 4874 . . . . . . . . 9 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ⟨(𝑎(.r𝑆)(1r𝑆)), (𝑏(.r𝑅)(1r𝑅))⟩ = ⟨𝑎, 𝑏⟩)
5953, 58eqtrd 2764 . . . . . . . 8 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨𝑎, 𝑏⟩(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = ⟨𝑎, 𝑏⟩)
60 oveq1 7409 . . . . . . . . 9 (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = (⟨𝑎, 𝑏⟩(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩))
6160, 44eqeq12d 2740 . . . . . . . 8 (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝑥(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = 𝑥 ↔ (⟨𝑎, 𝑏⟩(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = ⟨𝑎, 𝑏⟩))
6259, 61syl5ibrcom 246 . . . . . . 7 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = 𝑥))
6362rexlimdvva 3203 . . . . . 6 (𝜑 → (∃𝑎 ∈ (Base‘𝑆)∃𝑏 ∈ (Base‘𝑅)𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = 𝑥))
6423, 63biimtrid 241 . . . . 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 18597 . 2 (𝜑 → ⟨(1r𝑆), (1r𝑅)⟩ = (1r𝑌))
6867eqcomd 2730 1 (𝜑 → (1r𝑌) = ⟨(1r𝑆), (1r𝑅)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wrex 3062  cop 4627   × cxp 5665  cfv 6534  (class class class)co 7402  Basecbs 17149  .rcmulr 17203   ×s cxps 17457  mulGrpcmgp 20035  1rcur 20082  Ringcrg 20134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-tp 4626  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8700  df-map 8819  df-ixp 8889  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-sup 9434  df-inf 9435  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12472  df-z 12558  df-dec 12677  df-uz 12822  df-fz 13486  df-struct 17085  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-plusg 17215  df-mulr 17216  df-sca 17218  df-vsca 17219  df-ip 17220  df-tset 17221  df-ple 17222  df-ds 17224  df-hom 17226  df-cco 17227  df-0g 17392  df-prds 17398  df-imas 17459  df-xps 17461  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-mgp 20036  df-ur 20083  df-ring 20136
This theorem is referenced by:  rngqipring1  21165  pzriprng1  21374
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