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Theorem xpsring1d 20269
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 2728 . . . 4 (mulGrp‘𝑌) = (mulGrp‘𝑌)
2 eqid 2728 . . . 4 (Base‘𝑌) = (Base‘𝑌)
31, 2mgpbas 20080 . . 3 (Base‘𝑌) = (Base‘(mulGrp‘𝑌))
4 eqid 2728 . . . 4 (1r𝑌) = (1r𝑌)
51, 4ringidval 20123 . . 3 (1r𝑌) = (0g‘(mulGrp‘𝑌))
6 eqid 2728 . . . 4 (.r𝑌) = (.r𝑌)
71, 6mgpplusg 20078 . . 3 (.r𝑌) = (+g‘(mulGrp‘𝑌))
8 xpsringd.s . . . . . 6 (𝜑𝑆 ∈ Ring)
9 eqid 2728 . . . . . . 7 (Base‘𝑆) = (Base‘𝑆)
10 eqid 2728 . . . . . . 7 (1r𝑆) = (1r𝑆)
119, 10ringidcl 20202 . . . . . 6 (𝑆 ∈ Ring → (1r𝑆) ∈ (Base‘𝑆))
128, 11syl 17 . . . . 5 (𝜑 → (1r𝑆) ∈ (Base‘𝑆))
13 xpsringd.r . . . . . 6 (𝜑𝑅 ∈ Ring)
14 eqid 2728 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
15 eqid 2728 . . . . . . 7 (1r𝑅) = (1r𝑅)
1614, 15ringidcl 20202 . . . . . 6 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
1713, 16syl 17 . . . . 5 (𝜑 → (1r𝑅) ∈ (Base‘𝑅))
1812, 17opelxpd 5717 . . . 4 (𝜑 → ⟨(1r𝑆), (1r𝑅)⟩ ∈ ((Base‘𝑆) × (Base‘𝑅)))
19 xpsringd.y . . . . 5 𝑌 = (𝑆 ×s 𝑅)
2019, 9, 14, 8, 13xpsbas 17554 . . . 4 (𝜑 → ((Base‘𝑆) × (Base‘𝑅)) = (Base‘𝑌))
2118, 20eleqtrd 2831 . . 3 (𝜑 → ⟨(1r𝑆), (1r𝑅)⟩ ∈ (Base‘𝑌))
2220eleq2d 2815 . . . . 5 (𝜑 → (𝑥 ∈ ((Base‘𝑆) × (Base‘𝑅)) ↔ 𝑥 ∈ (Base‘𝑌)))
23 elxp2 5702 . . . . . 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 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 2728 . . . . . . . . . . 11 (.r𝑆) = (.r𝑆)
319, 30, 24, 26, 28ringcld 20199 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r𝑆)(.r𝑆)𝑎) ∈ (Base‘𝑆))
32 eqid 2728 . . . . . . . . . . 11 (.r𝑅) = (.r𝑅)
3314, 32, 25, 27, 29ringcld 20199 . . . . . . . . . 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 17557 . . . . . . . . 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 20205 . . . . . . . . . . 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 20205 . . . . . . . . . . 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 4882 . . . . . . . . 9 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ⟨((1r𝑆)(.r𝑆)𝑎), ((1r𝑅)(.r𝑅)𝑏)⟩ = ⟨𝑎, 𝑏⟩)
4234, 41eqtrd 2768 . . . . . . . 8 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨(1r𝑆), (1r𝑅)⟩(.r𝑌)⟨𝑎, 𝑏⟩) = ⟨𝑎, 𝑏⟩)
43 oveq2 7428 . . . . . . . . 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 2744 . . . . . . . 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 3208 . . . . . 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 20199 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(.r𝑆)(1r𝑆)) ∈ (Base‘𝑆))
5214, 32, 25, 29, 27ringcld 20199 . . . . . . . . . 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 17557 . . . . . . . . 9 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨𝑎, 𝑏⟩(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = ⟨(𝑎(.r𝑆)(1r𝑆)), (𝑏(.r𝑅)(1r𝑅))⟩)
549, 30, 10ringridm 20206 . . . . . . . . . . 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 20206 . . . . . . . . . . 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 4882 . . . . . . . . 9 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ⟨(𝑎(.r𝑆)(1r𝑆)), (𝑏(.r𝑅)(1r𝑅))⟩ = ⟨𝑎, 𝑏⟩)
5953, 58eqtrd 2768 . . . . . . . 8 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨𝑎, 𝑏⟩(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = ⟨𝑎, 𝑏⟩)
60 oveq1 7427 . . . . . . . . 9 (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = (⟨𝑎, 𝑏⟩(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩))
6160, 44eqeq12d 2744 . . . . . . . 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 3208 . . . . . 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 18628 . 2 (𝜑 → ⟨(1r𝑆), (1r𝑅)⟩ = (1r𝑌))
6867eqcomd 2734 1 (𝜑 → (1r𝑌) = ⟨(1r𝑆), (1r𝑅)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wrex 3067  cop 4635   × cxp 5676  cfv 6548  (class class class)co 7420  Basecbs 17180  .rcmulr 17234   ×s cxps 17488  mulGrpcmgp 20074  1rcur 20121  Ringcrg 20173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-er 8725  df-map 8847  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9466  df-inf 9467  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-3 12307  df-4 12308  df-5 12309  df-6 12310  df-7 12311  df-8 12312  df-9 12313  df-n0 12504  df-z 12590  df-dec 12709  df-uz 12854  df-fz 13518  df-struct 17116  df-sets 17133  df-slot 17151  df-ndx 17163  df-base 17181  df-plusg 17246  df-mulr 17247  df-sca 17249  df-vsca 17250  df-ip 17251  df-tset 17252  df-ple 17253  df-ds 17255  df-hom 17257  df-cco 17258  df-0g 17423  df-prds 17429  df-imas 17490  df-xps 17492  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-mgp 20075  df-ur 20122  df-ring 20175
This theorem is referenced by:  rngqipring1  21206  pzriprng1  21424
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