Theorem opifismgm 18695

Description: A structure with a group addition operation expressed by a conditional operator is a magma if both values of the conditional operator are contained in the base set. (Contributed by AV, 9-Feb-2020.)

Hypotheses

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

Assertion

Ref	Expression
opifismgm	⊢ (𝜑 → 𝑀 ∈ Mgm)

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

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

Proof of Theorem opifismgm

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

Step	Hyp	Ref	Expression
1		opifismgm.c	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐵)
2		opifismgm.d	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐷 ∈ 𝐵)
3	1, 2	ifcld 4529	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → if(𝜓, 𝐶, 𝐷) ∈ 𝐵)
4	3	ralrimivva 3207	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵)
5	4	adantr 484	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵)
6		simprl 780	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝐵)
7		simprr 782	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝐵)
8		opifismgm.p	. . . . 5 ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝜓, 𝐶, 𝐷))
9	8	ovmpoelrn 8055	. . . 4 ⊢ ((∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵 ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)
10	5, 6, 7, 9	syl3anc 1392	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)
11	10	ralrimivva 3207	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)
12		opifismgm.n	. . 3 ⊢ (𝜑 → 𝐵 ≠ ∅)
13		n0 4307	. . . 4 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐵)
14		opifismgm.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
15		eqid 2764	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
16	14, 15	ismgmn0 18678	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵))
17	16	exlimiv 1952	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵))
18	13, 17	sylbi 219	. . 3 ⊢ (𝐵 ≠ ∅ → (𝑀 ∈ Mgm ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵))
19	12, 18	syl 17	. 2 ⊢ (𝜑 → (𝑀 ∈ Mgm ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵))
20	11, 19	mpbird 259	1 ⊢ (𝜑 → 𝑀 ∈ Mgm)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562  ∃wex 1801   ∈ wcel 2144   ≠ wne 2959  ∀wral 3078  ∅c0 4287  ifcif 4482  ‘cfv 6523  (class class class)co 7398   ∈ cmpo 7400  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-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

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-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-1st 7972  df-2nd 7973  df-mgm 18676

This theorem is referenced by:  mgm2nsgrplem1  18957  sgrp2nmndlem1  18962