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Theorem rngosubdir 37476
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 2725 . . . . 5 (invβ€˜πΊ) = (invβ€˜πΊ)
4 ringsubdi.4 . . . . 5 𝐷 = ( /𝑔 β€˜πΊ)
51, 2, 3, 4rngosub 37460 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) = (𝐴𝐺((invβ€˜πΊ)β€˜π΅)))
653adant3r3 1181 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴𝐷𝐡) = (𝐴𝐺((invβ€˜πΊ)β€˜π΅)))
76oveq1d 7431 . 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 37431 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ (𝐴𝐻𝐢) ∈ 𝑋)
1093adant3r2 1180 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴𝐻𝐢) ∈ 𝑋)
111, 8, 2rngocl 37431 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ (𝐡𝐻𝐢) ∈ 𝑋)
12113adant3r1 1179 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐡𝐻𝐢) ∈ 𝑋)
1310, 12jca 510 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐻𝐢) ∈ 𝑋 ∧ (𝐡𝐻𝐢) ∈ 𝑋))
141, 2, 3, 4rngosub 37460 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐢) ∈ 𝑋 ∧ (𝐡𝐻𝐢) ∈ 𝑋) β†’ ((𝐴𝐻𝐢)𝐷(𝐡𝐻𝐢)) = ((𝐴𝐻𝐢)𝐺((invβ€˜πΊ)β€˜(𝐡𝐻𝐢))))
15143expb 1117 . . . 4 ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐢) ∈ 𝑋 ∧ (𝐡𝐻𝐢) ∈ 𝑋)) β†’ ((𝐴𝐻𝐢)𝐷(𝐡𝐻𝐢)) = ((𝐴𝐻𝐢)𝐺((invβ€˜πΊ)β€˜(𝐡𝐻𝐢))))
1613, 15syldan 589 . . 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 37457 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐡 ∈ 𝑋) β†’ ((invβ€˜πΊ)β€˜π΅) ∈ 𝑋)
1918ex 411 . . . . . . 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 1439 . . . . . 6 (𝑅 ∈ RingOps β†’ ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ (𝐴 ∈ 𝑋 ∧ ((invβ€˜πΊ)β€˜π΅) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)))
2221imp 405 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴 ∈ 𝑋 ∧ ((invβ€˜πΊ)β€˜π΅) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋))
231, 8, 2rngodir 37435 . . . . 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 589 . . . 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 37473 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ ((invβ€˜πΊ)β€˜(𝐡𝐻𝐢)) = (((invβ€˜πΊ)β€˜π΅)𝐻𝐢))
26253adant3r1 1179 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((invβ€˜πΊ)β€˜(𝐡𝐻𝐢)) = (((invβ€˜πΊ)β€˜π΅)𝐻𝐢))
2726oveq2d 7432 . . . 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 2768 . . 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 2768 . 2 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐻𝐢)𝐷(𝐡𝐻𝐢)) = ((𝐴𝐺((invβ€˜πΊ)β€˜π΅))𝐻𝐢))
307, 29eqtr4d 2768 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ran crn 5673  β€˜cfv 6543  (class class class)co 7416  1st c1st 7989  2nd c2nd 7990  invcgn 30345   /𝑔 cgs 30346  RingOpscrngo 37424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992  df-grpo 30347  df-gid 30348  df-ginv 30349  df-gdiv 30350  df-ablo 30399  df-ass 37373  df-exid 37375  df-mgmOLD 37379  df-sgrOLD 37391  df-mndo 37397  df-rngo 37425
This theorem is referenced by: (None)
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