Theorem ismndd 18809

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 1138	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
3		simpll 778	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝜑)
4		simplrl 788	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑥 ∈ 𝐵)
5		simplrr 789	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑦 ∈ 𝐵)
6		simpr 489	. . . . . . 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 1399	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
9	8	ralrimiva 3157	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
10	2, 9	jca 520	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))))
11	10	ralrimivva 3208	. . 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 7427	. . . . . . 7 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐺)𝑦))
15	14, 12	eleq12d 2857	. . . . . 6 ⊢ (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺)))
16		eqidd 2764	. . . . . . . . 9 ⊢ (𝜑 → 𝑧 = 𝑧)
17	13, 14, 16	oveq123d 7431	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 + 𝑦) + 𝑧) = ((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧))
18		eqidd 2764	. . . . . . . . 9 ⊢ (𝜑 → 𝑥 = 𝑥)
19	13	oveqd 7427	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 + 𝑧) = (𝑦(+g‘𝐺)𝑧))
20	13, 18, 19	oveq123d 7431	. . . . . . . 8 ⊢ (𝜑 → (𝑥 + (𝑦 + 𝑧)) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧)))
21	17, 20	eqeq12d 2779	. . . . . . 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 3338	. . . . . 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 643	. . . . 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 3338	. . . 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 3338	. . 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 235	. 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 2865	. . 3 ⊢ (𝜑 → 0 ∈ (Base‘𝐺))
29	12	eleq2d 2849	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐺)))
30	29	biimpar 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ 𝐵)
31	13	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → + = (+g‘𝐺))
32	31	oveqd 7427	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = ( 0 (+g‘𝐺)𝑥))
33		ismndd.i	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
34	32, 33	eqtr3d 2800	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 (+g‘𝐺)𝑥) = 𝑥)
35	31	oveqd 7427	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = (𝑥(+g‘𝐺) 0 ))
36		ismndd.j	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)
37	35, 36	eqtr3d 2800	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝐺) 0 ) = 𝑥)
38	34, 37	jca 520	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (( 0 (+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺) 0 ) = 𝑥))
39	30, 38	syldan 602	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (( 0 (+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺) 0 ) = 𝑥))
40	39	ralrimiva 3157	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐺)(( 0 (+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺) 0 ) = 𝑥))
41		oveq1 7417	. . . . . 6 ⊢ (𝑢 = 0 → (𝑢(+g‘𝐺)𝑥) = ( 0 (+g‘𝐺)𝑥))
42	41	eqeq1d 2765	. . . . 5 ⊢ (𝑢 = 0 → ((𝑢(+g‘𝐺)𝑥) = 𝑥 ↔ ( 0 (+g‘𝐺)𝑥) = 𝑥))
43	42	ovanraleqv 7434	. . . 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 3581	. . 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 595	. 2 ⊢ (𝜑 → ∃𝑢 ∈ (Base‘𝐺)∀𝑥 ∈ (Base‘𝐺)((𝑢(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑢) = 𝑥))
46		eqid 2763	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
47		eqid 2763	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
48	46, 47	ismnd 18790	. 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 594	1 ⊢ (𝜑 → 𝐺 ∈ Mnd)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  +_gcplusg 17305  Mndcmnd 18787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413  df-mgm 18693  df-sgrp 18772  df-mnd 18788

This theorem is referenced by:  issubmnd  18814  prdsmndd  18823  imasmnd2  18827  frmdmnd  18913  isgrpde  19019  oppgmnd  19419  isringd  20370  iscrngd  20371  xrsmcmn  21545  xrs1mnd  21590  rloccring  33591  idlsrgmnd  33804  bj-endmnd  37962  endmndlem  49793