Theorem opmpoismgm 47544

Description: A structure with a group addition operation in maps-to notation is a magma if the operation value is contained in the base set. (Contributed by AV, 16-Feb-2020.)

Hypotheses

Ref	Expression
opmpoismgm.b	⊢ 𝐵 = (Base‘𝑀)
opmpoismgm.p	⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)
opmpoismgm.n	⊢ (𝜑 → 𝐵 ≠ ∅)
opmpoismgm.c	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐵)

Assertion

Ref	Expression
opmpoismgm	⊢ (𝜑 → 𝑀 ∈ Mgm)

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

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝑀(𝑦)

Proof of Theorem opmpoismgm

Dummy variables 𝑎 𝑏 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opmpoismgm.c	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐵)
2	1	ralrimivva 3191	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐵)
3	2	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐵)
4		simprl 769	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝐵)
5		simprr 771	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝐵)
6		eqid 2726	. . . . 5 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)
7	6	ovmpoelrn 8086	. . . 4 ⊢ ((∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏) ∈ 𝐵)
8	3, 4, 5, 7	syl3anc 1368	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏) ∈ 𝐵)
9	8	ralrimivva 3191	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏) ∈ 𝐵)
10		opmpoismgm.n	. . 3 ⊢ (𝜑 → 𝐵 ≠ ∅)
11		n0 4349	. . . 4 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑒 𝑒 ∈ 𝐵)
12		opmpoismgm.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
13		opmpoismgm.p	. . . . . . 7 ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)
14	13	eqcomi 2735	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (+g‘𝑀)
15	12, 14	ismgmn0 18635	. . . . 5 ⊢ (𝑒 ∈ 𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏) ∈ 𝐵))
16	15	exlimiv 1926	. . . 4 ⊢ (∃𝑒 𝑒 ∈ 𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏) ∈ 𝐵))
17	11, 16	sylbi 216	. . 3 ⊢ (𝐵 ≠ ∅ → (𝑀 ∈ Mgm ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏) ∈ 𝐵))
18	10, 17	syl 17	. 2 ⊢ (𝜑 → (𝑀 ∈ Mgm ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏) ∈ 𝐵))
19	9, 18	mpbird 256	1 ⊢ (𝜑 → 𝑀 ∈ Mgm)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534  ∃wex 1774   ∈ wcel 2099   ≠ wne 2930  ∀wral 3051  ∅c0 4325  ‘cfv 6554  (class class class)co 7424   ∈ cmpo 7426  Basecbs 17213  +_gcplusg 17266  Mgmcmgm 18631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-mgm 18633

This theorem is referenced by:  copissgrp  47545