Theorem ismndd 18734

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 1120	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
3		simpll 766	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝜑)
4		simplrl 776	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑥 ∈ 𝐵)
5		simplrr 777	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑦 ∈ 𝐵)
6		simpr 484	. . . . . . 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 1374	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
9	8	ralrimiva 3132	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
10	2, 9	jca 511	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))))
11	10	ralrimivva 3187	. . 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 7422	. . . . . . 7 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐺)𝑦))
15	14, 12	eleq12d 2828	. . . . . 6 ⊢ (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺)))
16		eqidd 2736	. . . . . . . . 9 ⊢ (𝜑 → 𝑧 = 𝑧)
17	13, 14, 16	oveq123d 7426	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 + 𝑦) + 𝑧) = ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧))
18		eqidd 2736	. . . . . . . . 9 ⊢ (𝜑 → 𝑥 = 𝑥)
19	13	oveqd 7422	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 + 𝑧) = (𝑦(+g‘𝐺)𝑧))
20	13, 18, 19	oveq123d 7426	. . . . . . . 8 ⊢ (𝜑 → (𝑥 + (𝑦 + 𝑧)) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧)))
21	17, 20	eqeq12d 2751	. . . . . . 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 3325	. . . . . 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 632	. . . . 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 3325	. . . 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 3325	. . 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 232	. 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 2836	. . 3 ⊢ (𝜑 → 0 ∈ (Base‘𝐺))
29	12	eleq2d 2820	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐺)))
30	29	biimpar 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ 𝐵)
31	13	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → + = (+g‘𝐺))
32	31	oveqd 7422	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = ( 0 (+g‘𝐺)𝑥))
33		ismndd.i	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
34	32, 33	eqtr3d 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 (+g‘𝐺)𝑥) = 𝑥)
35	31	oveqd 7422	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = (𝑥(+g‘𝐺) 0 ))
36		ismndd.j	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)
37	35, 36	eqtr3d 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝐺) 0 ) = 𝑥)
38	34, 37	jca 511	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (( 0 (+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺) 0 ) = 𝑥))
39	30, 38	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (( 0 (+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺) 0 ) = 𝑥))
40	39	ralrimiva 3132	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐺)(( 0 (+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺) 0 ) = 𝑥))
41		oveq1 7412	. . . . . 6 ⊢ (𝑢 = 0 → (𝑢(+g‘𝐺)𝑥) = ( 0 (+g‘𝐺)𝑥))
42	41	eqeq1d 2737	. . . . 5 ⊢ (𝑢 = 0 → ((𝑢(+g‘𝐺)𝑥) = 𝑥 ↔ ( 0 (+g‘𝐺)𝑥) = 𝑥))
43	42	ovanraleqv 7429	. . . 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 3601	. . 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 584	. 2 ⊢ (𝜑 → ∃𝑢 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑢(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑢) = 𝑥))
46		eqid 2735	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
47		eqid 2735	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
48	46, 47	ismnd 18715	. 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 583	1 ⊢ (𝜑 → 𝐺 ∈ Mnd)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  +_gcplusg 17271  Mndcmnd 18712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6484  df-fv 6539  df-ov 7408  df-mgm 18618  df-sgrp 18697  df-mnd 18713

This theorem is referenced by:  issubmnd  18739  prdsmndd  18748  imasmnd2  18752  frmdmnd  18837  isgrpde  18940  oppgmnd  19337  isringd  20251  iscrngd  20252  xrsmcmn  21354  xrs1mnd  21372  rloccring  33265  idlsrgmnd  33529  bj-endmnd  37336  endmndlem  48990