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Theorem grpidpropdg 12811
Description: If two structures have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, they have the same identity element. (Contributed by Mario Carneiro, 27-Nov-2014.)
Hypotheses
Ref Expression
grpidpropd.1 (𝜑𝐵 = (Base‘𝐾))
grpidpropd.2 (𝜑𝐵 = (Base‘𝐿))
grpidproddg.k (𝜑𝐾𝑉)
grpidproddg.l (𝜑𝐿𝑊)
grpidpropd.3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
Assertion
Ref Expression
grpidpropdg (𝜑 → (0g𝐾) = (0g𝐿))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝜑,𝑥,𝑦   𝑥,𝐿,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem grpidpropdg
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpidpropd.3 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
21eqeq1d 2196 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥(+g𝐾)𝑦) = 𝑦 ↔ (𝑥(+g𝐿)𝑦) = 𝑦))
31oveqrspc2v 5915 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧(+g𝐾)𝑤) = (𝑧(+g𝐿)𝑤))
43oveqrspc2v 5915 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑥𝐵)) → (𝑦(+g𝐾)𝑥) = (𝑦(+g𝐿)𝑥))
54ancom2s 566 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑦(+g𝐾)𝑥) = (𝑦(+g𝐿)𝑥))
65eqeq1d 2196 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑦(+g𝐾)𝑥) = 𝑦 ↔ (𝑦(+g𝐿)𝑥) = 𝑦))
72, 6anbi12d 473 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦) ↔ ((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦)))
87anassrs 400 . . . . . 6 (((𝜑𝑥𝐵) ∧ 𝑦𝐵) → (((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦) ↔ ((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦)))
98ralbidva 2483 . . . . 5 ((𝜑𝑥𝐵) → (∀𝑦𝐵 ((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦) ↔ ∀𝑦𝐵 ((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦)))
109pm5.32da 452 . . . 4 (𝜑 → ((𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦)) ↔ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦))))
11 grpidpropd.1 . . . . . 6 (𝜑𝐵 = (Base‘𝐾))
1211eleq2d 2257 . . . . 5 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐾)))
1311raleqdv 2689 . . . . 5 (𝜑 → (∀𝑦𝐵 ((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦) ↔ ∀𝑦 ∈ (Base‘𝐾)((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦)))
1412, 13anbi12d 473 . . . 4 (𝜑 → ((𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦)) ↔ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦))))
15 grpidpropd.2 . . . . . 6 (𝜑𝐵 = (Base‘𝐿))
1615eleq2d 2257 . . . . 5 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐿)))
1715raleqdv 2689 . . . . 5 (𝜑 → (∀𝑦𝐵 ((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦) ↔ ∀𝑦 ∈ (Base‘𝐿)((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦)))
1816, 17anbi12d 473 . . . 4 (𝜑 → ((𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦)) ↔ (𝑥 ∈ (Base‘𝐿) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦))))
1910, 14, 183bitr3d 218 . . 3 (𝜑 → ((𝑥 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦)) ↔ (𝑥 ∈ (Base‘𝐿) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦))))
2019iotabidv 5211 . 2 (𝜑 → (℩𝑥(𝑥 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦))) = (℩𝑥(𝑥 ∈ (Base‘𝐿) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦))))
21 grpidproddg.k . . 3 (𝜑𝐾𝑉)
22 eqid 2187 . . . 4 (Base‘𝐾) = (Base‘𝐾)
23 eqid 2187 . . . 4 (+g𝐾) = (+g𝐾)
24 eqid 2187 . . . 4 (0g𝐾) = (0g𝐾)
2522, 23, 24grpidvalg 12810 . . 3 (𝐾𝑉 → (0g𝐾) = (℩𝑥(𝑥 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦))))
2621, 25syl 14 . 2 (𝜑 → (0g𝐾) = (℩𝑥(𝑥 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦))))
27 grpidproddg.l . . 3 (𝜑𝐿𝑊)
28 eqid 2187 . . . 4 (Base‘𝐿) = (Base‘𝐿)
29 eqid 2187 . . . 4 (+g𝐿) = (+g𝐿)
30 eqid 2187 . . . 4 (0g𝐿) = (0g𝐿)
3128, 29, 30grpidvalg 12810 . . 3 (𝐿𝑊 → (0g𝐿) = (℩𝑥(𝑥 ∈ (Base‘𝐿) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦))))
3227, 31syl 14 . 2 (𝜑 → (0g𝐿) = (℩𝑥(𝑥 ∈ (Base‘𝐿) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦))))
3320, 26, 323eqtr4d 2230 1 (𝜑 → (0g𝐾) = (0g𝐿))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1363  wcel 2158  wral 2465  cio 5188  cfv 5228  (class class class)co 5888  Basecbs 12475  +gcplusg 12550  0gc0g 12722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-riota 5844  df-ov 5891  df-inn 8933  df-ndx 12478  df-slot 12479  df-base 12481  df-0g 12724
This theorem is referenced by:  mhmpropd  12876  grppropd  12912  grpinvpropdg  12969  mulgpropdg  13054  rngidpropdg  13378  sralmod0g  13604
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