Theorem dedalg 10647

Description: A deductive system is an "algebra".

Assertion

Ref	Expression
dedalg	|- (T e. Ded -> T e. Alg)

Proof of Theorem dedalg

Step	Hyp	Ref	Expression
1		relded 10644	. . . . 5 |- Rel Ded
2		reldded 10645	. . . . 5 |- Rel dom Ded
3		relrded 10646	. . . . 5 |- Rel ran Ded
4	1, 2, 3	3pm3.2i 820	. . . 4 |- (Rel Ded /\ Rel dom Ded /\ Rel ran Ded)
5		11st22nd 10448	. . . 4 |- (((Rel Ded /\ Rel dom Ded /\ Rel ran Ded) /\ T e. Ded) -> T = <.<.(1st` (1st` T)), (2nd` (1st` T))>., <.(1st` (2nd` T)), (2nd` (2nd` T))>.>.)
6	4, 5	mpan 697	. . 3 |- (T e. Ded -> T = <.<.(1st` (1st` T)), (2nd` (1st` T))>., <.(1st` (2nd` T)), (2nd` (2nd` T))>.>.)
7		eqid 1478	. . . . . 6 |- (dom` T) = (dom` T)
8	7	domval 10626	. . . . 5 |- (dom` T) = (1st` (1st` T))
9		eqid 1478	. . . . . 6 |- (cod` T) = (cod` T)
10	9	codval 10627	. . . . 5 |- (cod` T) = (2nd` (1st` T))
11	8, 10	opeq12i 2496	. . . 4 |- <.(dom` T), (cod` T)>. = <.(1st` (1st` T)), (2nd` (1st` T))>.
12		eqid 1478	. . . . . 6 |- (id` T) = (id` T)
13	12	idval 10628	. . . . 5 |- (id` T) = (1st` (2nd` T))
14		eqid 1478	. . . . . 6 |- (o` T) = (o` T)
15	14	cmpval 10629	. . . . 5 |- (o` T) = (2nd` (2nd` T))
16	13, 15	opeq12i 2496	. . . 4 |- <.(id` T), (o` T)>. = <.(1st` (2nd` T)), (2nd` (2nd` T))>.
17	11, 16	opeq12i 2496	. . 3 |- <.<.(dom` T), (cod` T)>., <.(id` T), (o` T)>.>. = <.<.(1st` (1st` T)), (2nd` (1st` T))>., <.(1st` (2nd` T)), (2nd` (2nd` T))>.>.
18	6, 17	syl6eqr 1528	. 2 |- (T e. Ded -> T = <.<.(dom` T), (cod` T)>., <.(id` T), (o` T)>.>.)
19		eqid 1478	. . . . 5 |- dom (dom` T) = dom (dom` T)
20		eqid 1478	. . . . 5 |- dom (id` T) = dom (id` T)
21	7, 9, 12, 14, 19, 20	dedi 10641	. . . 4 |- (T e. Ded -> ((<.<.(dom` T), (cod` T)>., <.(id` T), (o` T)>.>. e. Alg /\ A.z e. dom (id` T)(((dom` T)` ((id` T)` z)) = z /\ ((cod` T)` ((id` T)` z)) = z) /\ A.x e. dom (dom` T)A.y e. dom (dom` T)(<.y, x>. e. dom (o` T) <-> ((dom` T)` y) = ((cod` T)` x))) /\ (A.x e. dom (dom` T)A.y e. dom (dom` T)(((dom` T)` y) = ((cod` T)` x) -> ((dom` T)` (y(o` T)x)) = ((dom` T)` x)) /\ A.x e. dom (dom` T)A.y e. dom (dom` T)(((dom` T)` y) = ((cod` T)` x) -> ((cod` T)` (y(o` T)x)) = ((cod` T)` y)))))
22	21	pm3.26d 321	. . 3 |- (T e. Ded -> (<.<.(dom` T), (cod` T)>., <.(id` T), (o` T)>.>. e. Alg /\ A.z e. dom (id` T)(((dom` T)` ((id` T)` z)) = z /\ ((cod` T)` ((id` T)` z)) = z) /\ A.x e. dom (dom` T)A.y e. dom (dom` T)(<.y, x>. e. dom (o` T) <-> ((dom` T)` y) = ((cod` T)` x))))
23	22	3simp1d 796	. 2 |- (T e. Ded -> <.<.(dom` T), (cod` T)>., <.(id` T), (o` T)>.>. e. Alg)
24	18, 23	eqeltrd 1551	1 |- (T e. Ded -> T e. Alg)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648  <.cop 2415  dom cdm 3176  ran crn 3177  Rel wrel 3181  ` cfv 3188  (class class class)co 3969  1stc1st 4083  2ndc2nd 4084  Algcalg 10614  domcdom_ 10615  codccod_ 10616  idcid_ 10617  oco_ 10618  Dedcded 10638

This theorem is referenced by:  rdmob 10652  rcmob 10653  aidm 10654  domc 10669  codc 10670  idc 10671  cmppfc 10672  mrdmcd 10693

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-int 2538  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-opr 3971  df-1st 4085  df-2nd 4086  df-doma 10620  df-coda 10621  df-ida 10622  df-cmpa 10623  df-ded 10639

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