Theorem ringdir 8143

Description: Distributive law for the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.)

Hypotheses

Ref	Expression
ringdi.1	⊢ G = (1st ‘R)
ringdi.2	⊢ H = (2nd ‘R)
ringdi.3	⊢ X = ran G

Assertion

Ref	Expression
ringdir	⊢ ((R ∈ Ring ⋀ (A ∈ X ⋀ B ∈ X ⋀ C ∈ X)) → ((AGB)HC) = ((AHC)G(BHC)))

Proof of Theorem ringdir

Step	Hyp	Ref	Expression
1		opreq1 3974	. . . . . . 7 ⊢ (x = A → (xGy) = (AGy))
2	1	opreq1d 3981	. . . . . 6 ⊢ (x = A → ((xGy)Hz) = ((AGy)Hz))
3		opreq1 3974	. . . . . . 7 ⊢ (x = A → (xHz) = (AHz))
4	3	opreq1d 3981	. . . . . 6 ⊢ (x = A → ((xHz)G(yHz)) = ((AHz)G(yHz)))
5	2, 4	eqeq12d 1492	. . . . 5 ⊢ (x = A → (((xGy)Hz) = ((xHz)G(yHz)) ↔ ((AGy)Hz) = ((AHz)G(yHz))))
6		opreq2 3975	. . . . . . 7 ⊢ (y = B → (AGy) = (AGB))
7	6	opreq1d 3981	. . . . . 6 ⊢ (y = B → ((AGy)Hz) = ((AGB)Hz))
8		opreq1 3974	. . . . . . 7 ⊢ (y = B → (yHz) = (BHz))
9	8	opreq2d 3982	. . . . . 6 ⊢ (y = B → ((AHz)G(yHz)) = ((AHz)G(BHz)))
10	7, 9	eqeq12d 1492	. . . . 5 ⊢ (y = B → (((AGy)Hz) = ((AHz)G(yHz)) ↔ ((AGB)Hz) = ((AHz)G(BHz))))
11		opreq2 3975	. . . . . 6 ⊢ (z = C → ((AGB)Hz) = ((AGB)HC))
12		opreq2 3975	. . . . . . 7 ⊢ (z = C → (AHz) = (AHC))
13		opreq2 3975	. . . . . . 7 ⊢ (z = C → (BHz) = (BHC))
14	12, 13	opreq12d 3984	. . . . . 6 ⊢ (z = C → ((AHz)G(BHz)) = ((AHC)G(BHC)))
15	11, 14	eqeq12d 1492	. . . . 5 ⊢ (z = C → (((AGB)Hz) = ((AHz)G(BHz)) ↔ ((AGB)HC) = ((AHC)G(BHC))))
16	5, 10, 15	rcla43v 1885	. . . 4 ⊢ ((A ∈ X ⋀ B ∈ X ⋀ C ∈ X) → (∀x ∈ X ∀y ∈ X ∀z ∈ X ((xGy)Hz) = ((xHz)G(yHz)) → ((AGB)HC) = ((AHC)G(BHC))))
17		3simp3 792	. . . . . . 7 ⊢ ((((xHy)Hz) = (xH(yHz)) ⋀ (xH(yGz)) = ((xHy)G(xHz)) ⋀ ((xGy)Hz) = ((xHz)G(yHz))) → ((xGy)Hz) = ((xHz)G(yHz)))
18	17	r19.20si 1709	. . . . . 6 ⊢ (∀z ∈ X (((xHy)Hz) = (xH(yHz)) ⋀ (xH(yGz)) = ((xHy)G(xHz)) ⋀ ((xGy)Hz) = ((xHz)G(yHz))) → ∀z ∈ X ((xGy)Hz) = ((xHz)G(yHz)))
19	18	r19.20si 1709	. . . . 5 ⊢ (∀y ∈ X ∀z ∈ X (((xHy)Hz) = (xH(yHz)) ⋀ (xH(yGz)) = ((xHy)G(xHz)) ⋀ ((xGy)Hz) = ((xHz)G(yHz))) → ∀y ∈ X ∀z ∈ X ((xGy)Hz) = ((xHz)G(yHz)))
20	19	r19.20si 1709	. . . 4 ⊢ (∀x ∈ X ∀y ∈ X ∀z ∈ X (((xHy)Hz) = (xH(yHz)) ⋀ (xH(yGz)) = ((xHy)G(xHz)) ⋀ ((xGy)Hz) = ((xHz)G(yHz))) → ∀x ∈ X ∀y ∈ X ∀z ∈ X ((xGy)Hz) = ((xHz)G(yHz)))
21	16, 20	syl5 21	. . 3 ⊢ ((A ∈ X ⋀ B ∈ X ⋀ C ∈ X) → (∀x ∈ X ∀y ∈ X ∀z ∈ X (((xHy)Hz) = (xH(yHz)) ⋀ (xH(yGz)) = ((xHy)G(xHz)) ⋀ ((xGy)Hz) = ((xHz)G(yHz))) → ((AGB)HC) = ((AHC)G(BHC))))
22		ringdi.1	. . . . . 6 ⊢ G = (1st ‘R)
23		ringdi.2	. . . . . 6 ⊢ H = (2nd ‘R)
24		ringdi.3	. . . . . 6 ⊢ X = ran G
25	22, 23, 24	ringi 8138	. . . . 5 ⊢ (R ∈ Ring → ((G ∈ Abel ⋀ H:(X × X)–→X) ⋀ (∀x ∈ X ∀y ∈ X ∀z ∈ X (((xHy)Hz) = (xH(yHz)) ⋀ (xH(yGz)) = ((xHy)G(xHz)) ⋀ ((xGy)Hz) = ((xHz)G(yHz))) ⋀ ∃x ∈ X ∀y ∈ X ((yHx) = y ⋀ (xHy) = y))))
26	25	pm3.27d 325	. . . 4 ⊢ (R ∈ Ring → (∀x ∈ X ∀y ∈ X ∀z ∈ X (((xHy)Hz) = (xH(yHz)) ⋀ (xH(yGz)) = ((xHy)G(xHz)) ⋀ ((xGy)Hz) = ((xHz)G(yHz))) ⋀ ∃x ∈ X ∀y ∈ X ((yHx) = y ⋀ (xHy) = y)))
27	26	pm3.26d 321	. . 3 ⊢ (R ∈ Ring → ∀x ∈ X ∀y ∈ X ∀z ∈ X (((xHy)Hz) = (xH(yHz)) ⋀ (xH(yGz)) = ((xHy)G(xHz)) ⋀ ((xGy)Hz) = ((xHz)G(yHz))))
28	21, 27	syl5 21	. 2 ⊢ ((A ∈ X ⋀ B ∈ X ⋀ C ∈ X) → (R ∈ Ring → ((AGB)HC) = ((AHC)G(BHC))))
29	28	impcom 351	1 ⊢ ((R ∈ Ring ⋀ (A ∈ X ⋀ B ∈ X ⋀ C ∈ X)) → ((AGB)HC) = ((AHC)G(BHC)))

Syntax hints:   → wi 3   ⋀ wa 223   ⋀ w3a 777   = wceq 958   ∈ wcel 960  ∀wral 1648  ∃wrex 1649   × cxp 3174  ran crn 3177  –→wf 3184   ‘cfv 3188  (class class class)co 3969  1^st c1st 4083  2^nd c2nd 4084  Abelcabl 8095  Ringcring 8135

This theorem is referenced by:  ring2 8145

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-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-fv 3204  df-opr 3971  df-1st 4085  df-2nd 4086  df-ring 8136

Copyright terms: Public domain