Theorem ismndd 13018

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 1206	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
3		simpll 527	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝜑)
4		simplrl 535	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑥 ∈ 𝐵)
5		simplrr 536	. . . . . . 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 1251	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
9	8	ralrimiva 2567	. . . . 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 2576	. . 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 5935	. . . . . . 7 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐺)𝑦))
15	14, 12	eleq12d 2264	. . . . . 6 ⊢ (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺)))
16		eqidd 2194	. . . . . . . . 9 ⊢ (𝜑 → 𝑧 = 𝑧)
17	13, 14, 16	oveq123d 5939	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 + 𝑦) + 𝑧) = ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧))
18		eqidd 2194	. . . . . . . . 9 ⊢ (𝜑 → 𝑥 = 𝑥)
19	13	oveqd 5935	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 + 𝑧) = (𝑦(+g‘𝐺)𝑧))
20	13, 18, 19	oveq123d 5939	. . . . . . . 8 ⊢ (𝜑 → (𝑥 + (𝑦 + 𝑧)) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧)))
21	17, 20	eqeq12d 2208	. . . . . . 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 2706	. . . . . 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 2706	. . . 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 2706	. . 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 2272	. . 3 ⊢ (𝜑 → 0 ∈ (Base‘𝐺))
29	12	eleq2d 2263	. . . . . 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 5935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = ( 0 (+g‘𝐺)𝑥))
33		ismndd.i	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
34	32, 33	eqtr3d 2228	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 (+g‘𝐺)𝑥) = 𝑥)
35	31	oveqd 5935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = (𝑥(+g‘𝐺) 0 ))
36		ismndd.j	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)
37	35, 36	eqtr3d 2228	. . . . . 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 2567	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐺)(( 0 (+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺) 0 ) = 𝑥))
41		oveq1 5925	. . . . . 6 ⊢ (𝑢 = 0 → (𝑢(+g‘𝐺)𝑥) = ( 0 (+g‘𝐺)𝑥))
42	41	eqeq1d 2202	. . . . 5 ⊢ (𝑢 = 0 → ((𝑢(+g‘𝐺)𝑥) = 𝑥 ↔ ( 0 (+g‘𝐺)𝑥) = 𝑥))
43	42	ovanraleqv 5942	. . . 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 2864	. . 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 2193	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
47		eqid 2193	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
48	46, 47	ismnd 13000	. 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 980   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  ‘cfv 5254  (class class class)co 5918  Basecbs 12618  +_gcplusg 12695  Mndcmnd 12997

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-ov 5921  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-mgm 12939  df-sgrp 12985  df-mnd 12998

This theorem is referenced by:  issubmnd  13023  isgrpde  13094  isringd  13537  iscrngd  13538