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Theorem xpsring1d 20347
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 2735 . . . 4 (mulGrp‘𝑌) = (mulGrp‘𝑌)
2 eqid 2735 . . . 4 (Base‘𝑌) = (Base‘𝑌)
31, 2mgpbas 20158 . . 3 (Base‘𝑌) = (Base‘(mulGrp‘𝑌))
4 eqid 2735 . . . 4 (1r𝑌) = (1r𝑌)
51, 4ringidval 20201 . . 3 (1r𝑌) = (0g‘(mulGrp‘𝑌))
6 eqid 2735 . . . 4 (.r𝑌) = (.r𝑌)
71, 6mgpplusg 20156 . . 3 (.r𝑌) = (+g‘(mulGrp‘𝑌))
8 xpsringd.s . . . . . 6 (𝜑𝑆 ∈ Ring)
9 eqid 2735 . . . . . . 7 (Base‘𝑆) = (Base‘𝑆)
10 eqid 2735 . . . . . . 7 (1r𝑆) = (1r𝑆)
119, 10ringidcl 20280 . . . . . 6 (𝑆 ∈ Ring → (1r𝑆) ∈ (Base‘𝑆))
128, 11syl 17 . . . . 5 (𝜑 → (1r𝑆) ∈ (Base‘𝑆))
13 xpsringd.r . . . . . 6 (𝜑𝑅 ∈ Ring)
14 eqid 2735 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
15 eqid 2735 . . . . . . 7 (1r𝑅) = (1r𝑅)
1614, 15ringidcl 20280 . . . . . 6 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
1713, 16syl 17 . . . . 5 (𝜑 → (1r𝑅) ∈ (Base‘𝑅))
1812, 17opelxpd 5728 . . . 4 (𝜑 → ⟨(1r𝑆), (1r𝑅)⟩ ∈ ((Base‘𝑆) × (Base‘𝑅)))
19 xpsringd.y . . . . 5 𝑌 = (𝑆 ×s 𝑅)
2019, 9, 14, 8, 13xpsbas 17619 . . . 4 (𝜑 → ((Base‘𝑆) × (Base‘𝑅)) = (Base‘𝑌))
2118, 20eleqtrd 2841 . . 3 (𝜑 → ⟨(1r𝑆), (1r𝑅)⟩ ∈ (Base‘𝑌))
2220eleq2d 2825 . . . . 5 (𝜑 → (𝑥 ∈ ((Base‘𝑆) × (Base‘𝑅)) ↔ 𝑥 ∈ (Base‘𝑌)))
23 elxp2 5713 . . . . . 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 2735 . . . . . . . . . . 11 (.r𝑆) = (.r𝑆)
319, 30, 24, 26, 28ringcld 20277 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ((1r𝑆)(.r𝑆)𝑎) ∈ (Base‘𝑆))
32 eqid 2735 . . . . . . . . . . 11 (.r𝑅) = (.r𝑅)
3314, 32, 25, 27, 29ringcld 20277 . . . . . . . . . 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 17622 . . . . . . . . 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 20283 . . . . . . . . . . 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 20283 . . . . . . . . . . 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 4886 . . . . . . . . 9 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ⟨((1r𝑆)(.r𝑆)𝑎), ((1r𝑅)(.r𝑅)𝑏)⟩ = ⟨𝑎, 𝑏⟩)
4234, 41eqtrd 2775 . . . . . . . 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 2751 . . . . . . . 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 3211 . . . . . 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 20277 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (𝑎(.r𝑆)(1r𝑆)) ∈ (Base‘𝑆))
5214, 32, 25, 29, 27ringcld 20277 . . . . . . . . . 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 17622 . . . . . . . . 9 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → (⟨𝑎, 𝑏⟩(.r𝑌)⟨(1r𝑆), (1r𝑅)⟩) = ⟨(𝑎(.r𝑆)(1r𝑆)), (𝑏(.r𝑅)(1r𝑅))⟩)
549, 30, 10ringridm 20284 . . . . . . . . . . 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 20284 . . . . . . . . . . 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 4886 . . . . . . . . 9 ((𝜑 ∧ (𝑎 ∈ (Base‘𝑆) ∧ 𝑏 ∈ (Base‘𝑅))) → ⟨(𝑎(.r𝑆)(1r𝑆)), (𝑏(.r𝑅)(1r𝑅))⟩ = ⟨𝑎, 𝑏⟩)
5953, 58eqtrd 2775 . . . . . . . 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 2751 . . . . . . . 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 3211 . . . . . 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 18694 . 2 (𝜑 → ⟨(1r𝑆), (1r𝑅)⟩ = (1r𝑌))
6867eqcomd 2741 1 (𝜑 → (1r𝑌) = ⟨(1r𝑆), (1r𝑅)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wrex 3068  cop 4637   × cxp 5687  cfv 6563  (class class class)co 7431  Basecbs 17245  .rcmulr 17299   ×s cxps 17553  mulGrpcmgp 20152  1rcur 20199  Ringcrg 20251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17488  df-prds 17494  df-imas 17555  df-xps 17557  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mgp 20153  df-ur 20200  df-ring 20253
This theorem is referenced by:  rngqipring1  21344  pzriprng1  21527
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