Theorem ismgm 18404

Description: The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)

Hypotheses

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

Assertion

Ref	Expression
ismgm	⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵))

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

Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem ismgm

Dummy variables 𝑏 𝑚 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6827	. . 3 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) ∈ V)
2		fveq2 6812	. . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
3		ismgm.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
4	2, 3	eqtr4di 2795	. . 3 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
5		fvexd 6827	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑚) ∈ V)
6		fveq2 6812	. . . . . 6 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
7	6	adantr 481	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑚) = (+g‘𝑀))
8		ismgm.o	. . . . 5 ⊢ ⚬ = (+g‘𝑀)
9	7, 8	eqtr4di 2795	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑚) = ⚬ )
10		simplr 766	. . . . 5 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → 𝑏 = 𝐵)
11		oveq 7323	. . . . . . . 8 ⊢ (𝑜 = ⚬ → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦))
12	11	adantl 482	. . . . . . 7 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦))
13	12, 10	eleq12d 2832	. . . . . 6 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → ((𝑥𝑜𝑦) ∈ 𝑏 ↔ (𝑥 ⚬ 𝑦) ∈ 𝐵))
14	10, 13	raleqbidv 3316	. . . . 5 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵))
15	10, 14	raleqbidv 3316	. . . 4 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵))
16	5, 9, 15	sbcied2 3773	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) → ([(+g‘𝑚) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵))
17	1, 4, 16	sbcied2 3773	. 2 ⊢ (𝑚 = 𝑀 → ([(Base‘𝑚) / 𝑏][(+g‘𝑚) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵))
18		df-mgm 18403	. 2 ⊢ Mgm = {𝑚 ∣ [(Base‘𝑚) / 𝑏][(+g‘𝑚) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏}
19	17, 18	elab2g 3621	1 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∀wral 3062  Vcvv 3441  [wsbc 3726  ‘cfv 6466  (class class class)co 7317  Basecbs 16989  +_gcplusg 17039  Mgmcmgm 18401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708  ax-nul 5245

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-ral 3063  df-rab 3405  df-v 3443  df-sbc 3727  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-br 5088  df-iota 6418  df-fv 6474  df-ov 7320  df-mgm 18403

This theorem is referenced by:  ismgmn0  18405  mgmcl  18406  issstrmgm  18414  mgm0  18417  issgrpv  18454  efmndmgm  18600  smndex1mgm  18622  0mgm  45593  ismgmd  45595  mgm2mgm  45686  lidlmmgm  45748