Theorem ismndd 13698

Description: Deduce a monoid from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
ismndd.b	⊢ (𝜑 → 𝐵 = (Base‘𝐺))
ismndd.p	⊢ (𝜑 → + = (+g‘𝐺))
ismndd.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
ismndd.a	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
ismndd.z	⊢ (𝜑 → 0 ∈ 𝐵)
ismndd.i	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
ismndd.j	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)

Assertion

Ref	Expression
ismndd	⊢ (𝜑 → 𝐺 ∈ Mnd)

Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝐺,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥, 0

Allowed substitution hints:   + (𝑥,𝑦,𝑧)   0 (𝑦,𝑧)

Proof of Theorem ismndd

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismndd.c	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
2	1	3expb 1231	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
3		simpll 527	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝜑)
4		simplrl 537	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑥 ∈ 𝐵)
5		simplrr 538	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑦 ∈ 𝐵)
6		simpr 110	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑧 ∈ 𝐵)
7		ismndd.a	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
8	3, 4, 5, 6, 7	syl13anc 1276	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
9	8	ralrimiva 2617	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
10	2, 9	jca 306	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))))
11	10	ralrimivva 2626	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))))
12		ismndd.b	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
13		ismndd.p	. . . . . . . 8 ⊢ (𝜑 → + = (+g‘𝐺))
14	13	oveqd 6075	. . . . . . 7 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐺)𝑦))
15	14, 12	eleq12d 2305	. . . . . 6 ⊢ (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺)))
16		eqidd 2235	. . . . . . . . 9 ⊢ (𝜑 → 𝑧 = 𝑧)
17	13, 14, 16	oveq123d 6079	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 + 𝑦) + 𝑧) = ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧))
18		eqidd 2235	. . . . . . . . 9 ⊢ (𝜑 → 𝑥 = 𝑥)
19	13	oveqd 6075	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 + 𝑧) = (𝑦(+g‘𝐺)𝑧))
20	13, 18, 19	oveq123d 6079	. . . . . . . 8 ⊢ (𝜑 → (𝑥 + (𝑦 + 𝑧)) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧)))
21	17, 20	eqeq12d 2249	. . . . . . 7 ⊢ (𝜑 → (((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ↔ ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))))
22	12, 21	raleqbidv 2759	. . . . . 6 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ↔ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))))
23	15, 22	anbi12d 473	. . . . 5 ⊢ (𝜑 → (((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ↔ ((𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧)))))
24	12, 23	raleqbidv 2759	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ↔ ∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧)))))
25	12, 24	raleqbidv 2759	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧)))))
26	11, 25	mpbid 147	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))))
27		ismndd.z	. . . 4 ⊢ (𝜑 → 0 ∈ 𝐵)
28	27, 12	eleqtrd 2313	. . 3 ⊢ (𝜑 → 0 ∈ (Base‘𝐺))
29	12	eleq2d 2304	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐺)))
30	29	biimpar 297	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ 𝐵)
31	13	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → + = (+g‘𝐺))
32	31	oveqd 6075	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = ( 0 (+g‘𝐺)𝑥))
33		ismndd.i	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
34	32, 33	eqtr3d 2269	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 (+g‘𝐺)𝑥) = 𝑥)
35	31	oveqd 6075	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = (𝑥(+g‘𝐺) 0 ))
36		ismndd.j	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)
37	35, 36	eqtr3d 2269	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝐺) 0 ) = 𝑥)
38	34, 37	jca 306	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (( 0 (+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺) 0 ) = 𝑥))
39	30, 38	syldan 282	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (( 0 (+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺) 0 ) = 𝑥))
40	39	ralrimiva 2617	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐺)(( 0 (+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺) 0 ) = 𝑥))
41		oveq1 6065	. . . . . 6 ⊢ (𝑢 = 0 → (𝑢(+g‘𝐺)𝑥) = ( 0 (+g‘𝐺)𝑥))
42	41	eqeq1d 2243	. . . . 5 ⊢ (𝑢 = 0 → ((𝑢(+g‘𝐺)𝑥) = 𝑥 ↔ ( 0 (+g‘𝐺)𝑥) = 𝑥))
43	42	ovanraleqv 6082	. . . 4 ⊢ (𝑢 = 0 → (∀𝑥 ∈ (Base‘𝐺)((𝑢(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑢) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝐺)(( 0 (+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺) 0 ) = 𝑥)))
44	43	rspcev 2923	. . 3 ⊢ (( 0 ∈ (Base‘𝐺) ∧ ∀𝑥 ∈ (Base‘𝐺)(( 0 (+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺) 0 ) = 𝑥)) → ∃𝑢 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑢(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑢) = 𝑥))
45	28, 40, 44	syl2anc 411	. 2 ⊢ (𝜑 → ∃𝑢 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑢(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑢) = 𝑥))
46		eqid 2234	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
47		eqid 2234	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
48	46, 47	ismnd 13680	. 2 ⊢ (𝐺 ∈ Mnd ↔ (∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺) ∧ ∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))) ∧ ∃𝑢 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑢(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑢) = 𝑥)))
49	26, 45, 48	sylanbrc 417	1 ⊢ (𝜑 → 𝐺 ∈ Mnd)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  ‘cfv 5357  (class class class)co 6058  Basecbs 13296  +_gcplusg 13374  Mndcmnd 13677

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-ov 6061  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-mgm 13619  df-sgrp 13665  df-mnd 13678

This theorem is referenced by:  issubmnd  13703  imasmnd2  13707  isgrpde  13777  prdsmndd  14136  isringd  14284  iscrngd  14285