Theorem issgrp 13105

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 12761	. . . . 5 ⊢ Base Fn V
2		vex 2766	. . . . 5 ⊢ 𝑔 ∈ V
3		funfvex 5578	. . . . . 6 ⊢ ((Fun Base ∧ 𝑔 ∈ dom Base) → (Base‘𝑔) ∈ V)
4	3	funfni 5361	. . . . 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 5561	. . . 4 ⊢ (𝑔 = 𝑀 → (Base‘𝑔) = (Base‘𝑀))
8		issgrp.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
9	7, 8	eqtr4di 2247	. . 3 ⊢ (𝑔 = 𝑀 → (Base‘𝑔) = 𝐵)
10		plusgslid 12815	. . . . . . 7 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
11	10	slotex 12730	. . . . . 6 ⊢ (𝑔 ∈ V → (+g‘𝑔) ∈ V)
12	11	elv 2767	. . . . 5 ⊢ (+g‘𝑔) ∈ V
13	12	a1i 9	. . . 4 ⊢ ((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑔) ∈ V)
14		fveq2 5561	. . . . . 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 2247	. . . 4 ⊢ ((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑔) = ⚬ )
18		simplr 528	. . . . 5 ⊢ (((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → 𝑏 = 𝐵)
19		id 19	. . . . . . . . . 10 ⊢ (𝑜 = ⚬ → 𝑜 = ⚬ )
20		oveq 5931	. . . . . . . . . 10 ⊢ (𝑜 = ⚬ → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦))
21		eqidd 2197	. . . . . . . . . 10 ⊢ (𝑜 = ⚬ → 𝑧 = 𝑧)
22	19, 20, 21	oveq123d 5946	. . . . . . . . 9 ⊢ (𝑜 = ⚬ → ((𝑥𝑜𝑦)𝑜𝑧) = ((𝑥 ⚬ 𝑦) ⚬ 𝑧))
23		eqidd 2197	. . . . . . . . . 10 ⊢ (𝑜 = ⚬ → 𝑥 = 𝑥)
24		oveq 5931	. . . . . . . . . 10 ⊢ (𝑜 = ⚬ → (𝑦𝑜𝑧) = (𝑦 ⚬ 𝑧))
25	19, 23, 24	oveq123d 5946	. . . . . . . . 9 ⊢ (𝑜 = ⚬ → (𝑥𝑜(𝑦𝑜𝑧)) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
26	22, 25	eqeq12d 2211	. . . . . . . 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 2709	. . . . . 6 ⊢ (((𝑔 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
29	18, 28	raleqbidv 2709	. . . . 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 2709	. . . 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 3027	. . 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 3027	. 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 13104	. 2 ⊢ Smgrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
34	32, 33	elrab2 2923	1 ⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ∀wral 2475  Vcvv 2763  [wsbc 2989   Fn wfn 5254  ‘cfv 5259  (class class class)co 5925  Basecbs 12703  +_gcplusg 12780  Mgmcmgm 13056  Smgrpcsgrp 13103

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-ov 5928  df-inn 9008  df-2 9066  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-sgrp 13104

This theorem is referenced by:  issgrpv  13106  issgrpn0  13107  isnsgrp  13108  sgrpmgm  13109  sgrpass  13110  sgrp0  13112  sgrp1  13113  rnglidlmsgrp  14129