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Theorem grpidpropdg 12685
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 2186 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥(+g𝐾)𝑦) = 𝑦 ↔ (𝑥(+g𝐿)𝑦) = 𝑦))
31oveqrspc2v 5896 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧(+g𝐾)𝑤) = (𝑧(+g𝐿)𝑤))
43oveqrspc2v 5896 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑥𝐵)) → (𝑦(+g𝐾)𝑥) = (𝑦(+g𝐿)𝑥))
54ancom2s 566 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑦(+g𝐾)𝑥) = (𝑦(+g𝐿)𝑥))
65eqeq1d 2186 . . . . . . . 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 2473 . . . . 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 2247 . . . . 5 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐾)))
1311raleqdv 2678 . . . . 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 2247 . . . . 5 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐿)))
1715raleqdv 2678 . . . . 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 5195 . 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 2177 . . . 4 (Base‘𝐾) = (Base‘𝐾)
23 eqid 2177 . . . 4 (+g𝐾) = (+g𝐾)
24 eqid 2177 . . . 4 (0g𝐾) = (0g𝐾)
2522, 23, 24grpidvalg 12684 . . 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 2177 . . . 4 (Base‘𝐿) = (Base‘𝐿)
29 eqid 2177 . . . 4 (+g𝐿) = (+g𝐿)
30 eqid 2177 . . . 4 (0g𝐿) = (0g𝐿)
3128, 29, 30grpidvalg 12684 . . 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 2220 1 (𝜑 → (0g𝐾) = (0g𝐿))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  cio 5172  cfv 5212  (class class class)co 5869  Basecbs 12445  +gcplusg 12518  0gc0g 12653
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-cnex 7893  ax-resscn 7894  ax-1re 7896  ax-addrcl 7899
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220  df-riota 5825  df-ov 5872  df-inn 8909  df-ndx 12448  df-slot 12449  df-base 12451  df-0g 12655
This theorem is referenced by:  mhmpropd  12747  grppropd  12783  grpinvpropdg  12834  mulgpropdg  12913  rngidpropdg  13138
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