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Theorem rngosubdir 37327
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 2726 . . . . 5 (invβ€˜πΊ) = (invβ€˜πΊ)
4 ringsubdi.4 . . . . 5 𝐷 = ( /𝑔 β€˜πΊ)
51, 2, 3, 4rngosub 37311 . . . 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 7420 . 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 37282 . . . . . 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 37282 . . . . . 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 511 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐻𝐢) ∈ 𝑋 ∧ (𝐡𝐻𝐢) ∈ 𝑋))
141, 2, 3, 4rngosub 37311 . . . . 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 590 . . 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 37308 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐡 ∈ 𝑋) β†’ ((invβ€˜πΊ)β€˜π΅) ∈ 𝑋)
1918ex 412 . . . . . . 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 406 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴 ∈ 𝑋 ∧ ((invβ€˜πΊ)β€˜π΅) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋))
231, 8, 2rngodir 37286 . . . . 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 590 . . . 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 37324 . . . . . 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 7421 . . . 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 2769 . . 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 2769 . 2 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐻𝐢)𝐷(𝐡𝐻𝐢)) = ((𝐴𝐺((invβ€˜πΊ)β€˜π΅))𝐻𝐢))
307, 29eqtr4d 2769 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ran crn 5670  β€˜cfv 6537  (class class class)co 7405  1st c1st 7972  2nd c2nd 7973  invcgn 30253   /𝑔 cgs 30254  RingOpscrngo 37275
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-grpo 30255  df-gid 30256  df-ginv 30257  df-gdiv 30258  df-ablo 30307  df-ass 37224  df-exid 37226  df-mgmOLD 37230  df-sgrOLD 37242  df-mndo 37248  df-rngo 37276
This theorem is referenced by: (None)
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