Theorem mgmpropd 18620

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 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝜑)
2		mgmpropd.k	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
3	2	eqcomd 2734	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝐾) = 𝐵)
4	3	eleq2d 2815	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↔ 𝑥 ∈ 𝐵))
5	4	biimpcd 248	. . . . . . . 8 ⊢ (𝑥 ∈ (Base‘𝐾) → (𝜑 → 𝑥 ∈ 𝐵))
6	5	adantr 479	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝜑 → 𝑥 ∈ 𝐵))
7	6	impcom 406	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → 𝑥 ∈ 𝐵)
8	3	eleq2d 2815	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ (Base‘𝐾) ↔ 𝑦 ∈ 𝐵))
9	8	biimpd 228	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (Base‘𝐾) → 𝑦 ∈ 𝐵))
10	9	adantld 489	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦 ∈ 𝐵))
11	10	imp 405	. . . . . 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 835	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
14	13	eleq1d 2814	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → ((𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐾)))
15	14	2ralbidva 3214	. . 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 2770	. . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
18	17	eleq2d 2815	. . . . 5 ⊢ (𝜑 → ((𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿)))
19	17, 18	raleqbidv 3340	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ (Base‘𝐾)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐾) ↔ ∀𝑦 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿)))
20	17, 19	raleqbidv 3340	. . 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 278	. 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 4350	. . . 4 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑎 𝑎 ∈ 𝐵)
24	2	eleq2d 2815	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ 𝐵 ↔ 𝑎 ∈ (Base‘𝐾)))
25		eqid 2728	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
26		eqid 2728	. . . . . . 7 ⊢ (+g‘𝐾) = (+g‘𝐾)
27	25, 26	ismgmn0 18611	. . . . . 6 ⊢ (𝑎 ∈ (Base‘𝐾) → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾)))
28	24, 27	biimtrdi 252	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 𝐵 → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾))))
29	28	exlimdv 1928	. . . 4 ⊢ (𝜑 → (∃𝑎 𝑎 ∈ 𝐵 → (𝐾 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾))))
30	23, 29	biimtrid 241	. . 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 2815	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ 𝐵 ↔ 𝑎 ∈ (Base‘𝐿)))
33		eqid 2728	. . . . . . 7 ⊢ (Base‘𝐿) = (Base‘𝐿)
34		eqid 2728	. . . . . . 7 ⊢ (+g‘𝐿) = (+g‘𝐿)
35	33, 34	ismgmn0 18611	. . . . . 6 ⊢ (𝑎 ∈ (Base‘𝐿) → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿)))
36	32, 35	biimtrdi 252	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 𝐵 → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿))))
37	36	exlimdv 1928	. . . 4 ⊢ (𝜑 → (∃𝑎 𝑎 ∈ 𝐵 → (𝐿 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐿)∀𝑦 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿))))
38	23, 37	biimtrid 241	. . 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 310	1 ⊢ (𝜑 → (𝐾 ∈ Mgm ↔ 𝐿 ∈ Mgm))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098   ≠ wne 2937  ∀wral 3058  ∅c0 4326  ‘cfv 6553  (class class class)co 7426  Basecbs 17189  +_gcplusg 17242  Mgmcmgm 18607

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 2166  ax-ext 2699  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-dm 5692  df-iota 6505  df-fv 6561  df-ov 7429  df-mgm 18609

This theorem is referenced by:  mgmhmpropd  18667