Theorem ismgm 18681

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 6937	. . 3 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) ∈ V)
2		fveq2 6922	. . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
3		ismgm.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
4	2, 3	eqtr4di 2798	. . 3 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
5		fvexd 6937	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑚) ∈ V)
6		fveq2 6922	. . . . . 6 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
7	6	adantr 480	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑚) = (+g‘𝑀))
8		ismgm.o	. . . . 5 ⊢ ⚬ = (+g‘𝑀)
9	7, 8	eqtr4di 2798	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑚) = ⚬ )
10		simplr 768	. . . . 5 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → 𝑏 = 𝐵)
11		oveq 7456	. . . . . . . 8 ⊢ (𝑜 = ⚬ → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦))
12	11	adantl 481	. . . . . . 7 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦))
13	12, 10	eleq12d 2838	. . . . . 6 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → ((𝑥𝑜𝑦) ∈ 𝑏 ↔ (𝑥 ⚬ 𝑦) ∈ 𝐵))
14	10, 13	raleqbidv 3354	. . . . 5 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵))
15	10, 14	raleqbidv 3354	. . . 4 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵))
16	5, 9, 15	sbcied2 3852	. . 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 3852	. 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 18680	. 2 ⊢ Mgm = {𝑚 ∣ [(Base‘𝑚) / 𝑏][(+g‘𝑚) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏}
19	17, 18	elab2g 3696	1 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488  [wsbc 3804  ‘cfv 6575  (class class class)co 7450  Basecbs 17260  +_gcplusg 17313  Mgmcmgm 18678

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-ext 2711  ax-nul 5324

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-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  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-iota 6527  df-fv 6583  df-ov 7453  df-mgm 18680

This theorem is referenced by:  ismgmn0  18682  mgmcl  18683  ismgmd  18692  issstrmgm  18693  mgm0  18696  issgrpv  18761  efmndmgm  18922  smndex1mgm  18944  rnglidlmmgm  21280  0mgm  47891  mgm2mgm  47952