Theorem ismgmn0 18680

Description: The predicate "is a magma" for a structure with a nonempty base set. (Contributed by AV, 29-Jan-2020.)

Hypotheses

Ref	Expression
ismgmn0.b	⊢ 𝐵 = (Base‘𝑀)
ismgmn0.o	⊢ ⚬ = (+g‘𝑀)

Assertion

Ref	Expression
ismgmn0	⊢ (𝐴 ∈ 𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑀,𝑦   𝑥, ⚬ ,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem ismgmn0

Step	Hyp	Ref	Expression
1		ismgmn0.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑀)
2	1	eleq2i 2836	. . . 4 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ (Base‘𝑀))
3	2	biimpi 216	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (Base‘𝑀))
4	3	elfvexd 6959	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝑀 ∈ V)
5		ismgmn0.o	. . 3 ⊢ ⚬ = (+g‘𝑀)
6	1, 5	ismgm 18679	. 2 ⊢ (𝑀 ∈ V → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵))
7	4, 6	syl 17	1 ⊢ (𝐴 ∈ 𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  Mgmcmgm 18676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-dm 5710  df-iota 6525  df-fv 6581  df-ov 7451  df-mgm 18678

This theorem is referenced by:  mgmpropd  18689  mgm1  18696  opifismgm  18697  issgrpn0  18760  xrsmgm  21442  opmpoismgm  47890  nnsgrpmgm  47899  2zrngamgm  47968  2zrngmmgm  47975