Theorem opmpoismgm 47340

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 2725	. . . . 5 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)
7	6	ovmpoelrn 8072	. . . 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 4342	. . . 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 2734	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (+g‘𝑀)
15	12, 14	ismgmn0 18599	. . . . 5 ⊢ (𝑒 ∈ 𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏) ∈ 𝐵))
16	15	exlimiv 1925	. . . 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 1533  ∃wex 1773   ∈ wcel 2098   ≠ wne 2930  ∀wral 3051  ∅c0 4318  ‘cfv 6542  (class class class)co 7415   ∈ cmpo 7417  Basecbs 17177  +_gcplusg 17230  Mgmcmgm 18595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7989  df-2nd 7990  df-mgm 18597

This theorem is referenced by:  copissgrp  47341