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Theorem abvmul 20716
Description: An absolute value distributes under multiplication. (Contributed by Mario Carneiro, 8-Sep-2014.)
Hypotheses
Ref Expression
abvf.a 𝐴 = (AbsValβ€˜π‘…)
abvf.b 𝐡 = (Baseβ€˜π‘…)
abvmul.t Β· = (.rβ€˜π‘…)
Assertion
Ref Expression
abvmul ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (πΉβ€˜(𝑋 Β· π‘Œ)) = ((πΉβ€˜π‘‹) Β· (πΉβ€˜π‘Œ)))

Proof of Theorem abvmul
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 abvf.a . . . . . . 7 𝐴 = (AbsValβ€˜π‘…)
21abvrcl 20708 . . . . . 6 (𝐹 ∈ 𝐴 β†’ 𝑅 ∈ Ring)
3 abvf.b . . . . . . 7 𝐡 = (Baseβ€˜π‘…)
4 eqid 2728 . . . . . . 7 (+gβ€˜π‘…) = (+gβ€˜π‘…)
5 abvmul.t . . . . . . 7 Β· = (.rβ€˜π‘…)
6 eqid 2728 . . . . . . 7 (0gβ€˜π‘…) = (0gβ€˜π‘…)
71, 3, 4, 5, 6isabv 20706 . . . . . 6 (𝑅 ∈ Ring β†’ (𝐹 ∈ 𝐴 ↔ (𝐹:𝐡⟢(0[,)+∞) ∧ βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘…)) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯(+gβ€˜π‘…)𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))))))
82, 7syl 17 . . . . 5 (𝐹 ∈ 𝐴 β†’ (𝐹 ∈ 𝐴 ↔ (𝐹:𝐡⟢(0[,)+∞) ∧ βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘…)) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯(+gβ€˜π‘…)𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))))))
98ibi 266 . . . 4 (𝐹 ∈ 𝐴 β†’ (𝐹:𝐡⟢(0[,)+∞) ∧ βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘…)) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯(+gβ€˜π‘…)𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦))))))
10 simpl 481 . . . . . . 7 (((πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯(+gβ€˜π‘…)𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦))) β†’ (πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)))
1110ralimi 3080 . . . . . 6 (βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯(+gβ€˜π‘…)𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦))) β†’ βˆ€π‘¦ ∈ 𝐡 (πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)))
1211adantl 480 . . . . 5 ((((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘…)) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯(+gβ€˜π‘…)𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))) β†’ βˆ€π‘¦ ∈ 𝐡 (πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)))
1312ralimi 3080 . . . 4 (βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘…)) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯(+gβ€˜π‘…)𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))) β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)))
149, 13simpl2im 502 . . 3 (𝐹 ∈ 𝐴 β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)))
15 fvoveq1 7449 . . . . 5 (π‘₯ = 𝑋 β†’ (πΉβ€˜(π‘₯ Β· 𝑦)) = (πΉβ€˜(𝑋 Β· 𝑦)))
16 fveq2 6902 . . . . . 6 (π‘₯ = 𝑋 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘‹))
1716oveq1d 7441 . . . . 5 (π‘₯ = 𝑋 β†’ ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) = ((πΉβ€˜π‘‹) Β· (πΉβ€˜π‘¦)))
1815, 17eqeq12d 2744 . . . 4 (π‘₯ = 𝑋 β†’ ((πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ↔ (πΉβ€˜(𝑋 Β· 𝑦)) = ((πΉβ€˜π‘‹) Β· (πΉβ€˜π‘¦))))
19 oveq2 7434 . . . . . 6 (𝑦 = π‘Œ β†’ (𝑋 Β· 𝑦) = (𝑋 Β· π‘Œ))
2019fveq2d 6906 . . . . 5 (𝑦 = π‘Œ β†’ (πΉβ€˜(𝑋 Β· 𝑦)) = (πΉβ€˜(𝑋 Β· π‘Œ)))
21 fveq2 6902 . . . . . 6 (𝑦 = π‘Œ β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘Œ))
2221oveq2d 7442 . . . . 5 (𝑦 = π‘Œ β†’ ((πΉβ€˜π‘‹) Β· (πΉβ€˜π‘¦)) = ((πΉβ€˜π‘‹) Β· (πΉβ€˜π‘Œ)))
2320, 22eqeq12d 2744 . . . 4 (𝑦 = π‘Œ β†’ ((πΉβ€˜(𝑋 Β· 𝑦)) = ((πΉβ€˜π‘‹) Β· (πΉβ€˜π‘¦)) ↔ (πΉβ€˜(𝑋 Β· π‘Œ)) = ((πΉβ€˜π‘‹) Β· (πΉβ€˜π‘Œ))))
2418, 23rspc2v 3622 . . 3 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) β†’ (πΉβ€˜(𝑋 Β· π‘Œ)) = ((πΉβ€˜π‘‹) Β· (πΉβ€˜π‘Œ))))
2514, 24syl5com 31 . 2 (𝐹 ∈ 𝐴 β†’ ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (πΉβ€˜(𝑋 Β· π‘Œ)) = ((πΉβ€˜π‘‹) Β· (πΉβ€˜π‘Œ))))
26253impib 1113 1 ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (πΉβ€˜(𝑋 Β· π‘Œ)) = ((πΉβ€˜π‘‹) Β· (πΉβ€˜π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058   class class class wbr 5152  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426  0cc0 11146   + caddc 11149   Β· cmul 11151  +∞cpnf 11283   ≀ cle 11287  [,)cico 13366  Basecbs 17187  +gcplusg 17240  .rcmulr 17241  0gc0g 17428  Ringcrg 20180  AbsValcabv 20703
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 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-map 8853  df-abv 20704
This theorem is referenced by:  abv1z  20719  abvneg  20721  abvrec  20723  abvdiv  20724  abvdom  20725  abvres  20726  nmmul  24601  sranlm  24621  abvcxp  27568  qabvexp  27579  ostthlem2  27581  ostth2lem2  27587  ostth3  27591
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