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	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
mgmpropd.l	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
mgmpropd.b	⊢ (𝜑 → 𝐵 ≠ ∅)
mgmpropd.p	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)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.

Step	Hyp	Ref	Expression
1		simpl 472	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝜑)
2		mgmpropd.k	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
3	2	eqcomd 2616	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝐾) = 𝐵)
4	3	eleq2d 2673	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↔ 𝑥 ∈ 𝐵))
5	4	biimpcd 238	. . . . . . . 8 ⊢ (𝑥 ∈ (Base‘𝐾) → (𝜑 → 𝑥 ∈ 𝐵))
6	5	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝜑 → 𝑥 ∈ 𝐵))
7	6	impcom 445	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥 ∈ 𝐵)
8	3	eleq2d 2673	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ (Base‘𝐾) ↔ 𝑦 ∈ 𝐵))
9	8	biimpd 218	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (Base‘𝐾) → 𝑦 ∈ 𝐵))
10	9	adantld 482	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦 ∈ 𝐵))
11	10	imp 444	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑦 ∈ 𝐵)
12		mgmpropd.p	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
13	1, 7, 11, 12	syl12anc 1316	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
14	13	eleq1d 2672	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐾)))
15	14	2ralbidva 2971	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾) ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐾)))
16		mgmpropd.l	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
17	2, 16	eqtr3d 2646	. . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
18	17	eleq2d 2673	. . . . 5 ⊢ (𝜑 → ((𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿)))
19	17, 18	raleqbidv 3129	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ (Base‘𝐾)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐾) ↔ ∀𝑦 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿)))
20	17, 19	raleqbidv 3129	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐾) ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿)))
21	15, 20	bitrd 267	. 2 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾) ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿)))
22		mgmpropd.b	. . 3 ⊢ (𝜑 → 𝐵 ≠ ∅)
23		n0 3890	. . . 4 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑎 𝑎 ∈ 𝐵)
24	2	eleq2d 2673	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ 𝐵 ↔ 𝑎 ∈ (Base‘𝐾)))
25		eqid 2610	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
26		eqid 2610	. . . . . . 7 ⊢ (+g‘𝐾) = (+g‘𝐾)
27	25, 26	ismgmn0 17067	. . . . . 6 ⊢ (𝑎 ∈ (Base‘𝐾) → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾)))
28	24, 27	syl6bi 242	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 𝐵 → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾))))
29	28	exlimdv 1848	. . . 4 ⊢ (𝜑 → (∃𝑎 𝑎 ∈ 𝐵 → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾))))
30	23, 29	syl5bi 231	. . 3 ⊢ (𝜑 → (𝐵 ≠ ∅ → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾))))
31	22, 30	mpd 15	. 2 ⊢ (𝜑 → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾)))
32	16	eleq2d 2673	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ 𝐵 ↔ 𝑎 ∈ (Base‘𝐿)))
33		eqid 2610	. . . . . . 7 ⊢ (Base‘𝐿) = (Base‘𝐿)
34		eqid 2610	. . . . . . 7 ⊢ (+g‘𝐿) = (+g‘𝐿)
35	33, 34	ismgmn0 17067	. . . . . 6 ⊢ (𝑎 ∈ (Base‘𝐿) → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿)))
36	32, 35	syl6bi 242	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 𝐵 → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿))))
37	36	exlimdv 1848	. . . 4 ⊢ (𝜑 → (∃𝑎 𝑎 ∈ 𝐵 → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿))))
38	23, 37	syl5bi 231	. . 3 ⊢ (𝜑 → (𝐵 ≠ ∅ → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿))))
39	22, 38	mpd 15	. 2 ⊢ (𝜑 → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿)))
40	21, 31, 39	3bitr4d 299	1 ⊢ (𝜑 → (𝐾 ∈ Mgm ↔ 𝐿 ∈ Mgm))

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