Theorem issgrp 13549

Description: The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)

Hypotheses

Ref	Expression
issgrp.b	⊢ 𝐵 = (Base‘𝑀)
issgrp.o	⊢ ⚬ = (+g‘𝑀)

Assertion

Ref	Expression
issgrp	⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))

Distinct variable groups:   𝑥,𝐵,𝑦,𝑧   𝑥,𝑀,𝑦,𝑧   𝑥, ⚬ ,𝑦,𝑧

Proof of Theorem issgrp

Dummy variables 𝑏 𝑔 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		basfn 13204	. . . . 5 ⊢ Base Fn V
2		vex 2806	. . . . 5 ⊢ 𝑔 ∈ V
3		funfvex 5665	. . . . . 6 ⊢ ((Fun Base ∧ 𝑔 ∈ dom Base) → (Base‘𝑔) ∈ V)
4	3	funfni 5439	. . . . 5 ⊢ ((Base Fn V ∧ 𝑔 ∈ V) → (Base‘𝑔) ∈ V)
5	1, 2, 4	mp2an 426	. . . 4 ⊢ (Base‘𝑔) ∈ V
6	5	a1i 9	. . 3 ⊢ (𝑔 = 𝑀 → (Base‘𝑔) ∈ V)
7		fveq2 5648	. . . 4 ⊢ (𝑔 = 𝑀 → (Base‘𝑔) = (Base‘𝑀))
8		issgrp.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
9	7, 8	eqtr4di 2282	. . 3 ⊢ (𝑔 = 𝑀 → (Base‘𝑔) = 𝐵)
10		plusgslid 13258	. . . . . . 7 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
11	10	slotex 13172	. . . . . 6 ⊢ (𝑔 ∈ V → (+g‘𝑔) ∈ V)
12	11	elv 2807	. . . . 5 ⊢ (+g‘𝑔) ∈ V
13	12	a1i 9	. . . 4 ⊢ ((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑔) ∈ V)
14		fveq2 5648	. . . . . 6 ⊢ (𝑔 = 𝑀 → (+g‘𝑔) = (+g‘𝑀))
15	14	adantr 276	. . . . 5 ⊢ ((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑔) = (+g‘𝑀))
16		issgrp.o	. . . . 5 ⊢ ⚬ = (+g‘𝑀)
17	15, 16	eqtr4di 2282	. . . 4 ⊢ ((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑔) = ⚬ )
18		simplr 529	. . . . 5 ⊢ (((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → 𝑏 = 𝐵)
19		id 19	. . . . . . . . . 10 ⊢ (𝑜 = ⚬ → 𝑜 = ⚬ )
20		oveq 6034	. . . . . . . . . 10 ⊢ (𝑜 = ⚬ → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦))
21		eqidd 2232	. . . . . . . . . 10 ⊢ (𝑜 = ⚬ → 𝑧 = 𝑧)
22	19, 20, 21	oveq123d 6049	. . . . . . . . 9 ⊢ (𝑜 = ⚬ → ((𝑥𝑜𝑦)𝑜𝑧) = ((𝑥 ⚬ 𝑦) ⚬ 𝑧))
23		eqidd 2232	. . . . . . . . . 10 ⊢ (𝑜 = ⚬ → 𝑥 = 𝑥)
24		oveq 6034	. . . . . . . . . 10 ⊢ (𝑜 = ⚬ → (𝑦𝑜𝑧) = (𝑦 ⚬ 𝑧))
25	19, 23, 24	oveq123d 6049	. . . . . . . . 9 ⊢ (𝑜 = ⚬ → (𝑥𝑜(𝑦𝑜𝑧)) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
26	22, 25	eqeq12d 2246	. . . . . . . 8 ⊢ (𝑜 = ⚬ → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
27	26	adantl 277	. . . . . . 7 ⊢ (((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
28	18, 27	raleqbidv 2747	. . . . . 6 ⊢ (((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
29	18, 28	raleqbidv 2747	. . . . 5 ⊢ (((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
30	18, 29	raleqbidv 2747	. . . 4 ⊢ (((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
31	13, 17, 30	sbcied2 3070	. . 3 ⊢ ((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) → ([(+g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
32	6, 9, 31	sbcied2 3070	. 2 ⊢ (𝑔 = 𝑀 → ([(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
33		df-sgrp 13548	. 2 ⊢ Smgrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
34	32, 33	elrab2 2966	1 ⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202  ∀wral 2511  Vcvv 2803  [wsbc 3032   Fn wfn 5328  ‘cfv 5333  (class class class)co 6028  Basecbs 13145  +_gcplusg 13223  Mgmcmgm 13500  Smgrpcsgrp 13547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-ov 6031  df-inn 9186  df-2 9244  df-ndx 13148  df-slot 13149  df-base 13151  df-plusg 13236  df-sgrp 13548

This theorem is referenced by:  issgrpv  13550  issgrpn0  13551  isnsgrp  13552  sgrpmgm  13553  sgrpass  13554  sgrp0  13556  sgrp1  13557  rnglidlmsgrp  14576