Theorem mgmpropd 45002

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 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝜑)
2		mgmpropd.k	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
3	2	eqcomd 2743	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝐾) = 𝐵)
4	3	eleq2d 2823	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↔ 𝑥 ∈ 𝐵))
5	4	biimpcd 252	. . . . . . . 8 ⊢ (𝑥 ∈ (Base‘𝐾) → (𝜑 → 𝑥 ∈ 𝐵))
6	5	adantr 484	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝜑 → 𝑥 ∈ 𝐵))
7	6	impcom 411	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥 ∈ 𝐵)
8	3	eleq2d 2823	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ (Base‘𝐾) ↔ 𝑦 ∈ 𝐵))
9	8	biimpd 232	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (Base‘𝐾) → 𝑦 ∈ 𝐵))
10	9	adantld 494	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦 ∈ 𝐵))
11	10	imp 410	. . . . . 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 837	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
14	13	eleq1d 2822	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐾)))
15	14	2ralbidva 3119	. . 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 2779	. . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
18	17	eleq2d 2823	. . . . 5 ⊢ (𝜑 → ((𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿)))
19	17, 18	raleqbidv 3313	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ (Base‘𝐾)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐾) ↔ ∀𝑦 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿)))
20	17, 19	raleqbidv 3313	. . 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 282	. 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 4261	. . . 4 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑎 𝑎 ∈ 𝐵)
24	2	eleq2d 2823	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ 𝐵 ↔ 𝑎 ∈ (Base‘𝐾)))
25		eqid 2737	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
26		eqid 2737	. . . . . . 7 ⊢ (+g‘𝐾) = (+g‘𝐾)
27	25, 26	ismgmn0 18116	. . . . . 6 ⊢ (𝑎 ∈ (Base‘𝐾) → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾)))
28	24, 27	syl6bi 256	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 𝐵 → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾))))
29	28	exlimdv 1941	. . . 4 ⊢ (𝜑 → (∃𝑎 𝑎 ∈ 𝐵 → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾))))
30	23, 29	syl5bi 245	. . 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 2823	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ 𝐵 ↔ 𝑎 ∈ (Base‘𝐿)))
33		eqid 2737	. . . . . . 7 ⊢ (Base‘𝐿) = (Base‘𝐿)
34		eqid 2737	. . . . . . 7 ⊢ (+g‘𝐿) = (+g‘𝐿)
35	33, 34	ismgmn0 18116	. . . . . 6 ⊢ (𝑎 ∈ (Base‘𝐿) → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿)))
36	32, 35	syl6bi 256	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 𝐵 → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿))))
37	36	exlimdv 1941	. . . 4 ⊢ (𝜑 → (∃𝑎 𝑎 ∈ 𝐵 → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿))))
38	23, 37	syl5bi 245	. . 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 314	1 ⊢ (𝜑 → (𝐾 ∈ Mgm ↔ 𝐿 ∈ Mgm))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543  ∃wex 1787   ∈ wcel 2110   ≠ wne 2940  ∀wral 3061  ∅c0 4237  ‘cfv 6380  (class class class)co 7213  Basecbs 16760  +_gcplusg 16802  Mgmcmgm 18112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-dm 5561  df-iota 6338  df-fv 6388  df-ov 7216  df-mgm 18114

This theorem is referenced by:  mgmhmpropd  45012