Theorem ismgmn0 18579

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 2829	. . . 4 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ (Base‘𝑀))
3	2	biimpi 216	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (Base‘𝑀))
4	3	elfvexd 6878	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝑀 ∈ V)
5		ismgmn0.o	. . 3 ⊢ ⚬ = (+g‘𝑀)
6	1, 5	ismgm 18578	. 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 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3442  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  Mgmcmgm 18575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-dm 5642  df-iota 6456  df-fv 6508  df-ov 7371  df-mgm 18577

This theorem is referenced by:  mgmpropd  18588  mgm1  18595  opifismgm  18596  issgrpn0  18659  xrsmgm  21373  opmpoismgm  48527  nnsgrpmgm  48536  2zrngamgm  48605  2zrngmmgm  48612