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Theorem mgmpropd 42569
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 475 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝜑)
2 mgmpropd.k . . . . . . . . . . 11 (𝜑𝐵 = (Base‘𝐾))
32eqcomd 2806 . . . . . . . . . 10 (𝜑 → (Base‘𝐾) = 𝐵)
43eleq2d 2865 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (Base‘𝐾) ↔ 𝑥𝐵))
54biimpcd 241 . . . . . . . 8 (𝑥 ∈ (Base‘𝐾) → (𝜑𝑥𝐵))
65adantr 473 . . . . . . 7 ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝜑𝑥𝐵))
76impcom 397 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥𝐵)
83eleq2d 2865 . . . . . . . . 9 (𝜑 → (𝑦 ∈ (Base‘𝐾) ↔ 𝑦𝐵))
98biimpd 221 . . . . . . . 8 (𝜑 → (𝑦 ∈ (Base‘𝐾) → 𝑦𝐵))
109adantld 485 . . . . . . 7 (𝜑 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦𝐵))
1110imp 396 . . . . . 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 866 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
1413eleq1d 2864 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾) ↔ (𝑥(+g𝐿)𝑦) ∈ (Base‘𝐾)))
15142ralbidva 3170 . . 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 2836 . . . 4 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
1817eleq2d 2865 . . . . 5 (𝜑 → ((𝑥(+g𝐿)𝑦) ∈ (Base‘𝐾) ↔ (𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿)))
1917, 18raleqbidv 3336 . . . 4 (𝜑 → (∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐾) ↔ ∀𝑦 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿)))
2017, 19raleqbidv 3336 . . 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 271 . 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 4132 . . . 4 (𝐵 ≠ ∅ ↔ ∃𝑎 𝑎𝐵)
242eleq2d 2865 . . . . . 6 (𝜑 → (𝑎𝐵𝑎 ∈ (Base‘𝐾)))
25 eqid 2800 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
26 eqid 2800 . . . . . . 7 (+g𝐾) = (+g𝐾)
2725, 26ismgmn0 17558 . . . . . 6 (𝑎 ∈ (Base‘𝐾) → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾)))
2824, 27syl6bi 245 . . . . 5 (𝜑 → (𝑎𝐵 → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾))))
2928exlimdv 2029 . . . 4 (𝜑 → (∃𝑎 𝑎𝐵 → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾))))
3023, 29syl5bi 234 . . 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 2865 . . . . . 6 (𝜑 → (𝑎𝐵𝑎 ∈ (Base‘𝐿)))
33 eqid 2800 . . . . . . 7 (Base‘𝐿) = (Base‘𝐿)
34 eqid 2800 . . . . . . 7 (+g𝐿) = (+g𝐿)
3533, 34ismgmn0 17558 . . . . . 6 (𝑎 ∈ (Base‘𝐿) → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿)))
3632, 35syl6bi 245 . . . . 5 (𝜑 → (𝑎𝐵 → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿))))
3736exlimdv 2029 . . . 4 (𝜑 → (∃𝑎 𝑎𝐵 → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿))))
3823, 37syl5bi 234 . . 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 303 1 (𝜑 → (𝐾 ∈ Mgm ↔ 𝐿 ∈ Mgm))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wex 1875  wcel 2157  wne 2972  wral 3090  c0 4116  cfv 6102  (class class class)co 6879  Basecbs 16183  +gcplusg 16266  Mgmcmgm 17554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-nul 4984  ax-pow 5036
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-dm 5323  df-iota 6065  df-fv 6110  df-ov 6882  df-mgm 17556
This theorem is referenced by:  mgmhmpropd  42579
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