Theorem opifismgm 17466

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 4271	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → if(𝜓, 𝐶, 𝐷) ∈ 𝐵)
4	3	ralrimivva 3120	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵)
5	4	adantr 466	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵)
6		simprl 754	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝐵)
7		simprr 756	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝐵)
8		opifismgm.p	. . . . 5 ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝜓, 𝐶, 𝐷))
9	8	ovmpt2elrn 7395	. . . 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 1476	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)
11	10	ralrimivva 3120	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)
12		opifismgm.n	. . 3 ⊢ (𝜑 → 𝐵 ≠ ∅)
13		n0 4079	. . . 4 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐵)
14		opifismgm.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
15		eqid 2771	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
16	14, 15	ismgmn0 17452	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵))
17	16	exlimiv 2010	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵))
18	13, 17	sylbi 207	. . 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 247	1 ⊢ (𝜑 → 𝑀 ∈ Mgm)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631  ∃wex 1852   ∈ wcel 2145   ≠ wne 2943  ∀wral 3061  ∅c0 4063  ifcif 4226  ‘cfv 6030  (class class class)co 6796   ↦ cmpt2 6798  Basecbs 16064  +_gcplusg 16149  Mgmcmgm 17448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-mgm 17450

This theorem is referenced by:  mgm2nsgrplem1  17613  sgrp2nmndlem1  17618