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Theorem rngosubdir 36809
Description: Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
ringsubdi.1 𝐺 = (1st β€˜π‘…)
ringsubdi.2 𝐻 = (2nd β€˜π‘…)
ringsubdi.3 𝑋 = ran 𝐺
ringsubdi.4 𝐷 = ( /𝑔 β€˜πΊ)
Assertion
Ref Expression
rngosubdir ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐷𝐡)𝐻𝐢) = ((𝐴𝐻𝐢)𝐷(𝐡𝐻𝐢)))
Proof of Theorem rngosubdir
StepHypRef Expression
1 ringsubdi.1 . . . . 5 𝐺 = (1st β€˜π‘…)
2 ringsubdi.3 . . . . 5 𝑋 = ran 𝐺
3 eqid 2732 . . . . 5 (invβ€˜πΊ) = (invβ€˜πΊ)
4 ringsubdi.4 . . . . 5 𝐷 = ( /𝑔 β€˜πΊ)
51, 2, 3, 4rngosub 36793 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) = (𝐴𝐺((invβ€˜πΊ)β€˜π΅)))
653adant3r3 1184 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴𝐷𝐡) = (𝐴𝐺((invβ€˜πΊ)β€˜π΅)))
76oveq1d 7423 . 2 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐷𝐡)𝐻𝐢) = ((𝐴𝐺((invβ€˜πΊ)β€˜π΅))𝐻𝐢))
8 ringsubdi.2 . . . . . . 7 𝐻 = (2nd β€˜π‘…)
91, 8, 2rngocl 36764 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ (𝐴𝐻𝐢) ∈ 𝑋)
1093adant3r2 1183 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴𝐻𝐢) ∈ 𝑋)
111, 8, 2rngocl 36764 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ (𝐡𝐻𝐢) ∈ 𝑋)
12113adant3r1 1182 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐡𝐻𝐢) ∈ 𝑋)
1310, 12jca 512 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐻𝐢) ∈ 𝑋 ∧ (𝐡𝐻𝐢) ∈ 𝑋))
141, 2, 3, 4rngosub 36793 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐢) ∈ 𝑋 ∧ (𝐡𝐻𝐢) ∈ 𝑋) β†’ ((𝐴𝐻𝐢)𝐷(𝐡𝐻𝐢)) = ((𝐴𝐻𝐢)𝐺((invβ€˜πΊ)β€˜(𝐡𝐻𝐢))))
15143expb 1120 . . . 4 ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐢) ∈ 𝑋 ∧ (𝐡𝐻𝐢) ∈ 𝑋)) β†’ ((𝐴𝐻𝐢)𝐷(𝐡𝐻𝐢)) = ((𝐴𝐻𝐢)𝐺((invβ€˜πΊ)β€˜(𝐡𝐻𝐢))))
1613, 15syldan 591 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐻𝐢)𝐷(𝐡𝐻𝐢)) = ((𝐴𝐻𝐢)𝐺((invβ€˜πΊ)β€˜(𝐡𝐻𝐢))))
17 idd 24 . . . . . . 7 (𝑅 ∈ RingOps β†’ (𝐴 ∈ 𝑋 β†’ 𝐴 ∈ 𝑋))
181, 2, 3rngonegcl 36790 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐡 ∈ 𝑋) β†’ ((invβ€˜πΊ)β€˜π΅) ∈ 𝑋)
1918ex 413 . . . . . . 7 (𝑅 ∈ RingOps β†’ (𝐡 ∈ 𝑋 β†’ ((invβ€˜πΊ)β€˜π΅) ∈ 𝑋))
20 idd 24 . . . . . . 7 (𝑅 ∈ RingOps β†’ (𝐢 ∈ 𝑋 β†’ 𝐢 ∈ 𝑋))
2117, 19, 203anim123d 1443 . . . . . 6 (𝑅 ∈ RingOps β†’ ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ (𝐴 ∈ 𝑋 ∧ ((invβ€˜πΊ)β€˜π΅) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)))
2221imp 407 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴 ∈ 𝑋 ∧ ((invβ€˜πΊ)β€˜π΅) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋))
231, 8, 2rngodir 36768 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ ((invβ€˜πΊ)β€˜π΅) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐺((invβ€˜πΊ)β€˜π΅))𝐻𝐢) = ((𝐴𝐻𝐢)𝐺(((invβ€˜πΊ)β€˜π΅)𝐻𝐢)))
2422, 23syldan 591 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐺((invβ€˜πΊ)β€˜π΅))𝐻𝐢) = ((𝐴𝐻𝐢)𝐺(((invβ€˜πΊ)β€˜π΅)𝐻𝐢)))
251, 8, 2, 3rngoneglmul 36806 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ ((invβ€˜πΊ)β€˜(𝐡𝐻𝐢)) = (((invβ€˜πΊ)β€˜π΅)𝐻𝐢))
26253adant3r1 1182 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((invβ€˜πΊ)β€˜(𝐡𝐻𝐢)) = (((invβ€˜πΊ)β€˜π΅)𝐻𝐢))
2726oveq2d 7424 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐻𝐢)𝐺((invβ€˜πΊ)β€˜(𝐡𝐻𝐢))) = ((𝐴𝐻𝐢)𝐺(((invβ€˜πΊ)β€˜π΅)𝐻𝐢)))
2824, 27eqtr4d 2775 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐺((invβ€˜πΊ)β€˜π΅))𝐻𝐢) = ((𝐴𝐻𝐢)𝐺((invβ€˜πΊ)β€˜(𝐡𝐻𝐢))))
2916, 28eqtr4d 2775 . 2 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐻𝐢)𝐷(𝐡𝐻𝐢)) = ((𝐴𝐺((invβ€˜πΊ)β€˜π΅))𝐻𝐢))
307, 29eqtr4d 2775 1 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐷𝐡)𝐻𝐢) = ((𝐴𝐻𝐢)𝐷(𝐡𝐻𝐢)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ran crn 5677  β€˜cfv 6543  (class class class)co 7408  1st c1st 7972  2nd c2nd 7973  invcgn 29739   /𝑔 cgs 29740  RingOpscrngo 36757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-grpo 29741  df-gid 29742  df-ginv 29743  df-gdiv 29744  df-ablo 29793  df-ass 36706  df-exid 36708  df-mgmOLD 36712  df-sgrOLD 36724  df-mndo 36730  df-rngo 36758
This theorem is referenced by: (None)
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