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Theorem rngosubdir 36455
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 2733 . . . . 5 (invβ€˜πΊ) = (invβ€˜πΊ)
4 ringsubdi.4 . . . . 5 𝐷 = ( /𝑔 β€˜πΊ)
51, 2, 3, 4rngosub 36439 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) = (𝐴𝐺((invβ€˜πΊ)β€˜π΅)))
653adant3r3 1185 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴𝐷𝐡) = (𝐴𝐺((invβ€˜πΊ)β€˜π΅)))
76oveq1d 7376 . 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 36410 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ (𝐴𝐻𝐢) ∈ 𝑋)
1093adant3r2 1184 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴𝐻𝐢) ∈ 𝑋)
111, 8, 2rngocl 36410 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ (𝐡𝐻𝐢) ∈ 𝑋)
12113adant3r1 1183 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐡𝐻𝐢) ∈ 𝑋)
1310, 12jca 513 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐻𝐢) ∈ 𝑋 ∧ (𝐡𝐻𝐢) ∈ 𝑋))
141, 2, 3, 4rngosub 36439 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐢) ∈ 𝑋 ∧ (𝐡𝐻𝐢) ∈ 𝑋) β†’ ((𝐴𝐻𝐢)𝐷(𝐡𝐻𝐢)) = ((𝐴𝐻𝐢)𝐺((invβ€˜πΊ)β€˜(𝐡𝐻𝐢))))
15143expb 1121 . . . 4 ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐢) ∈ 𝑋 ∧ (𝐡𝐻𝐢) ∈ 𝑋)) β†’ ((𝐴𝐻𝐢)𝐷(𝐡𝐻𝐢)) = ((𝐴𝐻𝐢)𝐺((invβ€˜πΊ)β€˜(𝐡𝐻𝐢))))
1613, 15syldan 592 . . 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 36436 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐡 ∈ 𝑋) β†’ ((invβ€˜πΊ)β€˜π΅) ∈ 𝑋)
1918ex 414 . . . . . . 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 1444 . . . . . 6 (𝑅 ∈ RingOps β†’ ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ (𝐴 ∈ 𝑋 ∧ ((invβ€˜πΊ)β€˜π΅) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)))
2221imp 408 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴 ∈ 𝑋 ∧ ((invβ€˜πΊ)β€˜π΅) ∈ 𝑋 ∧ 𝐢 ∈ 𝑋))
231, 8, 2rngodir 36414 . . . . 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 592 . . . 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 36452 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ ((invβ€˜πΊ)β€˜(𝐡𝐻𝐢)) = (((invβ€˜πΊ)β€˜π΅)𝐻𝐢))
26253adant3r1 1183 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((invβ€˜πΊ)β€˜(𝐡𝐻𝐢)) = (((invβ€˜πΊ)β€˜π΅)𝐻𝐢))
2726oveq2d 7377 . . . 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 2776 . . 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 2776 . 2 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴𝐻𝐢)𝐷(𝐡𝐻𝐢)) = ((𝐴𝐺((invβ€˜πΊ)β€˜π΅))𝐻𝐢))
307, 29eqtr4d 2776 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ran crn 5638  β€˜cfv 6500  (class class class)co 7361  1st c1st 7923  2nd c2nd 7924  invcgn 29482   /𝑔 cgs 29483  RingOpscrngo 36403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-grpo 29484  df-gid 29485  df-ginv 29486  df-gdiv 29487  df-ablo 29536  df-ass 36352  df-exid 36354  df-mgmOLD 36358  df-sgrOLD 36370  df-mndo 36376  df-rngo 36404
This theorem is referenced by: (None)
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