Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mgmpropd Structured version   Visualization version   GIF version

Theorem mgmpropd 41565
 Description: If two structures have the same (nonempty) base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a magma iff the other one is. (Contributed by AV, 25-Feb-2020.)
Hypotheses
Ref Expression
mgmpropd.k (𝜑𝐵 = (Base‘𝐾))
mgmpropd.l (𝜑𝐵 = (Base‘𝐿))
mgmpropd.b (𝜑𝐵 ≠ ∅)
mgmpropd.p ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
Assertion
Ref Expression
mgmpropd (𝜑 → (𝐾 ∈ Mgm ↔ 𝐿 ∈ Mgm))
Distinct variable groups:   𝑥,𝑦,𝐾   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem mgmpropd
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 simpl 472 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝜑)
2 mgmpropd.k . . . . . . . . . . 11 (𝜑𝐵 = (Base‘𝐾))
32eqcomd 2616 . . . . . . . . . 10 (𝜑 → (Base‘𝐾) = 𝐵)
43eleq2d 2673 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (Base‘𝐾) ↔ 𝑥𝐵))
54biimpcd 238 . . . . . . . 8 (𝑥 ∈ (Base‘𝐾) → (𝜑𝑥𝐵))
65adantr 480 . . . . . . 7 ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝜑𝑥𝐵))
76impcom 445 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥𝐵)
83eleq2d 2673 . . . . . . . . 9 (𝜑 → (𝑦 ∈ (Base‘𝐾) ↔ 𝑦𝐵))
98biimpd 218 . . . . . . . 8 (𝜑 → (𝑦 ∈ (Base‘𝐾) → 𝑦𝐵))
109adantld 482 . . . . . . 7 (𝜑 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦𝐵))
1110imp 444 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑦𝐵)
12 mgmpropd.p . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
131, 7, 11, 12syl12anc 1316 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
1413eleq1d 2672 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾) ↔ (𝑥(+g𝐿)𝑦) ∈ (Base‘𝐾)))
15142ralbidva 2971 . . 3 (𝜑 → (∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾) ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐾)))
16 mgmpropd.l . . . . 5 (𝜑𝐵 = (Base‘𝐿))
172, 16eqtr3d 2646 . . . 4 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
1817eleq2d 2673 . . . . 5 (𝜑 → ((𝑥(+g𝐿)𝑦) ∈ (Base‘𝐾) ↔ (𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿)))
1917, 18raleqbidv 3129 . . . 4 (𝜑 → (∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐾) ↔ ∀𝑦 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿)))
2017, 19raleqbidv 3129 . . 3 (𝜑 → (∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐾) ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿)))
2115, 20bitrd 267 . 2 (𝜑 → (∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾) ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿)))
22 mgmpropd.b . . 3 (𝜑𝐵 ≠ ∅)
23 n0 3890 . . . 4 (𝐵 ≠ ∅ ↔ ∃𝑎 𝑎𝐵)
242eleq2d 2673 . . . . . 6 (𝜑 → (𝑎𝐵𝑎 ∈ (Base‘𝐾)))
25 eqid 2610 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
26 eqid 2610 . . . . . . 7 (+g𝐾) = (+g𝐾)
2725, 26ismgmn0 17067 . . . . . 6 (𝑎 ∈ (Base‘𝐾) → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾)))
2824, 27syl6bi 242 . . . . 5 (𝜑 → (𝑎𝐵 → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾))))
2928exlimdv 1848 . . . 4 (𝜑 → (∃𝑎 𝑎𝐵 → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾))))
3023, 29syl5bi 231 . . 3 (𝜑 → (𝐵 ≠ ∅ → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾))))
3122, 30mpd 15 . 2 (𝜑 → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾)))
3216eleq2d 2673 . . . . . 6 (𝜑 → (𝑎𝐵𝑎 ∈ (Base‘𝐿)))
33 eqid 2610 . . . . . . 7 (Base‘𝐿) = (Base‘𝐿)
34 eqid 2610 . . . . . . 7 (+g𝐿) = (+g𝐿)
3533, 34ismgmn0 17067 . . . . . 6 (𝑎 ∈ (Base‘𝐿) → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿)))
3632, 35syl6bi 242 . . . . 5 (𝜑 → (𝑎𝐵 → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿))))
3736exlimdv 1848 . . . 4 (𝜑 → (∃𝑎 𝑎𝐵 → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿))))
3823, 37syl5bi 231 . . 3 (𝜑 → (𝐵 ≠ ∅ → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿))))
3922, 38mpd 15 . 2 (𝜑 → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿)))
4021, 31, 393bitr4d 299 1 (𝜑 → (𝐾 ∈ Mgm ↔ 𝐿 ∈ Mgm))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∅c0 3874  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  Mgmcmgm 17063 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717  ax-pow 4769 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-dm 5048  df-iota 5768  df-fv 5812  df-ov 6552  df-mgm 17065 This theorem is referenced by:  mgmhmpropd  41575
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