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Theorem mgmpropd 18584
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 482 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝜑)
2 mgmpropd.k . . . . . . . . . . 11 (𝜑𝐵 = (Base‘𝐾))
32eqcomd 2732 . . . . . . . . . 10 (𝜑 → (Base‘𝐾) = 𝐵)
43eleq2d 2813 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (Base‘𝐾) ↔ 𝑥𝐵))
54biimpcd 248 . . . . . . . 8 (𝑥 ∈ (Base‘𝐾) → (𝜑𝑥𝐵))
65adantr 480 . . . . . . 7 ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝜑𝑥𝐵))
76impcom 407 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥𝐵)
83eleq2d 2813 . . . . . . . . 9 (𝜑 → (𝑦 ∈ (Base‘𝐾) ↔ 𝑦𝐵))
98biimpd 228 . . . . . . . 8 (𝜑 → (𝑦 ∈ (Base‘𝐾) → 𝑦𝐵))
109adantld 490 . . . . . . 7 (𝜑 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦𝐵))
1110imp 406 . . . . . 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 834 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
1413eleq1d 2812 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾) ↔ (𝑥(+g𝐿)𝑦) ∈ (Base‘𝐾)))
15142ralbidva 3210 . . 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 2768 . . . 4 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
1817eleq2d 2813 . . . . 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 279 . 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 4341 . . . 4 (𝐵 ≠ ∅ ↔ ∃𝑎 𝑎𝐵)
242eleq2d 2813 . . . . . 6 (𝜑 → (𝑎𝐵𝑎 ∈ (Base‘𝐾)))
25 eqid 2726 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
26 eqid 2726 . . . . . . 7 (+g𝐾) = (+g𝐾)
2725, 26ismgmn0 18575 . . . . . 6 (𝑎 ∈ (Base‘𝐾) → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾)))
2824, 27biimtrdi 252 . . . . 5 (𝜑 → (𝑎𝐵 → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾))))
2928exlimdv 1928 . . . 4 (𝜑 → (∃𝑎 𝑎𝐵 → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) ∈ (Base‘𝐾))))
3023, 29biimtrid 241 . . 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 2813 . . . . . 6 (𝜑 → (𝑎𝐵𝑎 ∈ (Base‘𝐿)))
33 eqid 2726 . . . . . . 7 (Base‘𝐿) = (Base‘𝐿)
34 eqid 2726 . . . . . . 7 (+g𝐿) = (+g𝐿)
3533, 34ismgmn0 18575 . . . . . 6 (𝑎 ∈ (Base‘𝐿) → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿)))
3632, 35biimtrdi 252 . . . . 5 (𝜑 → (𝑎𝐵 → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿))))
3736exlimdv 1928 . . . 4 (𝜑 → (∃𝑎 𝑎𝐵 → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) ∈ (Base‘𝐿))))
3823, 37biimtrid 241 . . 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 311 1 (𝜑 → (𝐾 ∈ Mgm ↔ 𝐿 ∈ Mgm))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wex 1773  wcel 2098  wne 2934  wral 3055  c0 4317  cfv 6537  (class class class)co 7405  Basecbs 17153  +gcplusg 17206  Mgmcmgm 18571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-dm 5679  df-iota 6489  df-fv 6545  df-ov 7408  df-mgm 18573
This theorem is referenced by:  mgmhmpropd  18631
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