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Theorem grpidpropdg 12816
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 2198 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥(+g𝐾)𝑦) = 𝑦 ↔ (𝑥(+g𝐿)𝑦) = 𝑦))
31oveqrspc2v 5918 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧(+g𝐾)𝑤) = (𝑧(+g𝐿)𝑤))
43oveqrspc2v 5918 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑥𝐵)) → (𝑦(+g𝐾)𝑥) = (𝑦(+g𝐿)𝑥))
54ancom2s 566 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑦(+g𝐾)𝑥) = (𝑦(+g𝐿)𝑥))
65eqeq1d 2198 . . . . . . . 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 2486 . . . . 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 2259 . . . . 5 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐾)))
1311raleqdv 2692 . . . . 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 2259 . . . . 5 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐿)))
1715raleqdv 2692 . . . . 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 5214 . 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 2189 . . . 4 (Base‘𝐾) = (Base‘𝐾)
23 eqid 2189 . . . 4 (+g𝐾) = (+g𝐾)
24 eqid 2189 . . . 4 (0g𝐾) = (0g𝐾)
2522, 23, 24grpidvalg 12815 . . 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 2189 . . . 4 (Base‘𝐿) = (Base‘𝐿)
29 eqid 2189 . . . 4 (+g𝐿) = (+g𝐿)
30 eqid 2189 . . . 4 (0g𝐿) = (0g𝐿)
3128, 29, 30grpidvalg 12815 . . 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 2232 1 (𝜑 → (0g𝐾) = (0g𝐿))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160  wral 2468  cio 5191  cfv 5231  (class class class)co 5891  Basecbs 12480  +gcplusg 12555  0gc0g 12727
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-cnex 7920  ax-resscn 7921  ax-1re 7923  ax-addrcl 7926
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-iota 5193  df-fun 5233  df-fn 5234  df-fv 5239  df-riota 5847  df-ov 5894  df-inn 8938  df-ndx 12483  df-slot 12484  df-base 12486  df-0g 12729
This theorem is referenced by:  mhmpropd  12884  grppropd  12928  grpinvpropdg  12985  mulgpropdg  13070  rngidpropdg  13457  sralmod0g  13728
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