Theorem ismgm 18677

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 6884	. . 3 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) ∈ V)
2		fveq2 6869	. . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
3		ismgm.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
4	2, 3	eqtr4di 2817	. . 3 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
5		fvexd 6884	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑚) ∈ V)
6		fveq2 6869	. . . . . 6 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
7	6	adantr 484	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑚) = (+g‘𝑀))
8		ismgm.o	. . . . 5 ⊢ ⚬ = (+g‘𝑀)
9	7, 8	eqtr4di 2817	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑚) = ⚬ )
10		simplr 778	. . . . 5 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → 𝑏 = 𝐵)
11		oveq 7404	. . . . . . . 8 ⊢ (𝑜 = ⚬ → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦))
12	11	adantl 485	. . . . . . 7 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦))
13	12, 10	eleq12d 2858	. . . . . 6 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → ((𝑥𝑜𝑦) ∈ 𝑏 ↔ (𝑥 ⚬ 𝑦) ∈ 𝐵))
14	10, 13	raleqbidv 3338	. . . . 5 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵))
15	10, 14	raleqbidv 3338	. . . 4 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵))
16	5, 9, 15	sbcied2 3790	. . 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 3790	. 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 18676	. 2 ⊢ Mgm = {𝑚 ∣ [(Base‘𝑚) / 𝑏][(+g‘𝑚) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏}
19	17, 18	elab2g 3641	1 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078  Vcvv 3456  [wsbc 3746  ‘cfv 6523  (class class class)co 7398  Basecbs 17247  +_gcplusg 17288  Mgmcmgm 18674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-nul 5258

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-iota 6479  df-fv 6531  df-ov 7401  df-mgm 18676

This theorem is referenced by:  ismgmn0  18678  mgmcl  18679  ismgmd  18688  issstrmgm  18689  mgm0  18692  issgrpv  18757  efmndmgm  18921  smndex1mgm  18946  rnglidlmmgm  21317  0mgm  48793  mgm2mgm  48854