Users' Mathboxes 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 43546
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 483 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝜑)
2 mgmpropd.k . . . . . . . . . . 11 (𝜑𝐵 = (Base‘𝐾))
32eqcomd 2803 . . . . . . . . . 10 (𝜑 → (Base‘𝐾) = 𝐵)
43eleq2d 2870 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (Base‘𝐾) ↔ 𝑥𝐵))
54biimpcd 250 . . . . . . . 8 (𝑥 ∈ (Base‘𝐾) → (𝜑𝑥𝐵))
65adantr 481 . . . . . . 7 ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝜑𝑥𝐵))
76impcom 408 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥𝐵)
83eleq2d 2870 . . . . . . . . 9 (𝜑 → (𝑦 ∈ (Base‘𝐾) ↔ 𝑦𝐵))
98biimpd 230 . . . . . . . 8 (𝜑 → (𝑦 ∈ (Base‘𝐾) → 𝑦𝐵))
109adantld 491 . . . . . . 7 (𝜑 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦𝐵))
1110imp 407 . . . . . 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 833 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
1413eleq1d 2869 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾) ↔ (𝑥(+g𝐿)𝑦) ∈ (Base‘𝐾)))
15142ralbidva 3167 . . 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 2835 . . . 4 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
1817eleq2d 2870 . . . . 5 (𝜑 → ((𝑥(+g𝐿)𝑦) ∈ (Base‘𝐾) ↔ (𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿)))
1917, 18raleqbidv 3363 . . . 4 (𝜑 → (∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐾) ↔ ∀𝑦 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿)))
2017, 19raleqbidv 3363 . . 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 280 . 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 4236 . . . 4 (𝐵 ≠ ∅ ↔ ∃𝑎 𝑎𝐵)
242eleq2d 2870 . . . . . 6 (𝜑 → (𝑎𝐵𝑎 ∈ (Base‘𝐾)))
25 eqid 2797 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
26 eqid 2797 . . . . . . 7 (+g𝐾) = (+g𝐾)
2725, 26ismgmn0 17687 . . . . . 6 (𝑎 ∈ (Base‘𝐾) → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾)))
2824, 27syl6bi 254 . . . . 5 (𝜑 → (𝑎𝐵 → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾))))
2928exlimdv 1915 . . . 4 (𝜑 → (∃𝑎 𝑎𝐵 → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾))))
3023, 29syl5bi 243 . . 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 2870 . . . . . 6 (𝜑 → (𝑎𝐵𝑎 ∈ (Base‘𝐿)))
33 eqid 2797 . . . . . . 7 (Base‘𝐿) = (Base‘𝐿)
34 eqid 2797 . . . . . . 7 (+g𝐿) = (+g𝐿)
3533, 34ismgmn0 17687 . . . . . 6 (𝑎 ∈ (Base‘𝐿) → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿)))
3632, 35syl6bi 254 . . . . 5 (𝜑 → (𝑎𝐵 → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿))))
3736exlimdv 1915 . . . 4 (𝜑 → (∃𝑎 𝑎𝐵 → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿))))
3823, 37syl5bi 243 . . 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 312 1 (𝜑 → (𝐾 ∈ Mgm ↔ 𝐿 ∈ Mgm))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1525  wex 1765  wcel 2083  wne 2986  wral 3107  c0 4217  cfv 6232  (class class class)co 7023  Basecbs 16316  +gcplusg 16398  Mgmcmgm 17683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771  ax-nul 5108  ax-pow 5164
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-dm 5460  df-iota 6196  df-fv 6240  df-ov 7026  df-mgm 17685
This theorem is referenced by:  mgmhmpropd  43556
  Copyright terms: Public domain W3C validator