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Theorem lnfnl 31184
Description: Basic linearity property of a linear functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
lnfnl (((𝑇 ∈ LinFn ∧ 𝐴 ∈ β„‚) ∧ (𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹)) β†’ (π‘‡β€˜((𝐴 Β·β„Ž 𝐡) +β„Ž 𝐢)) = ((𝐴 Β· (π‘‡β€˜π΅)) + (π‘‡β€˜πΆ)))
Proof of Theorem lnfnl
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ellnfn 31136 . . . . . 6 (𝑇 ∈ LinFn ↔ (𝑇: β„‹βŸΆβ„‚ ∧ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ β„‹ βˆ€π‘§ ∈ β„‹ (π‘‡β€˜((π‘₯ Β·β„Ž 𝑦) +β„Ž 𝑧)) = ((π‘₯ Β· (π‘‡β€˜π‘¦)) + (π‘‡β€˜π‘§))))
21simprbi 498 . . . . 5 (𝑇 ∈ LinFn β†’ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ β„‹ βˆ€π‘§ ∈ β„‹ (π‘‡β€˜((π‘₯ Β·β„Ž 𝑦) +β„Ž 𝑧)) = ((π‘₯ Β· (π‘‡β€˜π‘¦)) + (π‘‡β€˜π‘§)))
3 oveq1 7416 . . . . . . . 8 (π‘₯ = 𝐴 β†’ (π‘₯ Β·β„Ž 𝑦) = (𝐴 Β·β„Ž 𝑦))
43fvoveq1d 7431 . . . . . . 7 (π‘₯ = 𝐴 β†’ (π‘‡β€˜((π‘₯ Β·β„Ž 𝑦) +β„Ž 𝑧)) = (π‘‡β€˜((𝐴 Β·β„Ž 𝑦) +β„Ž 𝑧)))
5 oveq1 7416 . . . . . . . 8 (π‘₯ = 𝐴 β†’ (π‘₯ Β· (π‘‡β€˜π‘¦)) = (𝐴 Β· (π‘‡β€˜π‘¦)))
65oveq1d 7424 . . . . . . 7 (π‘₯ = 𝐴 β†’ ((π‘₯ Β· (π‘‡β€˜π‘¦)) + (π‘‡β€˜π‘§)) = ((𝐴 Β· (π‘‡β€˜π‘¦)) + (π‘‡β€˜π‘§)))
74, 6eqeq12d 2749 . . . . . 6 (π‘₯ = 𝐴 β†’ ((π‘‡β€˜((π‘₯ Β·β„Ž 𝑦) +β„Ž 𝑧)) = ((π‘₯ Β· (π‘‡β€˜π‘¦)) + (π‘‡β€˜π‘§)) ↔ (π‘‡β€˜((𝐴 Β·β„Ž 𝑦) +β„Ž 𝑧)) = ((𝐴 Β· (π‘‡β€˜π‘¦)) + (π‘‡β€˜π‘§))))
8 oveq2 7417 . . . . . . . 8 (𝑦 = 𝐡 β†’ (𝐴 Β·β„Ž 𝑦) = (𝐴 Β·β„Ž 𝐡))
98fvoveq1d 7431 . . . . . . 7 (𝑦 = 𝐡 β†’ (π‘‡β€˜((𝐴 Β·β„Ž 𝑦) +β„Ž 𝑧)) = (π‘‡β€˜((𝐴 Β·β„Ž 𝐡) +β„Ž 𝑧)))
10 fveq2 6892 . . . . . . . . 9 (𝑦 = 𝐡 β†’ (π‘‡β€˜π‘¦) = (π‘‡β€˜π΅))
1110oveq2d 7425 . . . . . . . 8 (𝑦 = 𝐡 β†’ (𝐴 Β· (π‘‡β€˜π‘¦)) = (𝐴 Β· (π‘‡β€˜π΅)))
1211oveq1d 7424 . . . . . . 7 (𝑦 = 𝐡 β†’ ((𝐴 Β· (π‘‡β€˜π‘¦)) + (π‘‡β€˜π‘§)) = ((𝐴 Β· (π‘‡β€˜π΅)) + (π‘‡β€˜π‘§)))
139, 12eqeq12d 2749 . . . . . 6 (𝑦 = 𝐡 β†’ ((π‘‡β€˜((𝐴 Β·β„Ž 𝑦) +β„Ž 𝑧)) = ((𝐴 Β· (π‘‡β€˜π‘¦)) + (π‘‡β€˜π‘§)) ↔ (π‘‡β€˜((𝐴 Β·β„Ž 𝐡) +β„Ž 𝑧)) = ((𝐴 Β· (π‘‡β€˜π΅)) + (π‘‡β€˜π‘§))))
14 oveq2 7417 . . . . . . . 8 (𝑧 = 𝐢 β†’ ((𝐴 Β·β„Ž 𝐡) +β„Ž 𝑧) = ((𝐴 Β·β„Ž 𝐡) +β„Ž 𝐢))
1514fveq2d 6896 . . . . . . 7 (𝑧 = 𝐢 β†’ (π‘‡β€˜((𝐴 Β·β„Ž 𝐡) +β„Ž 𝑧)) = (π‘‡β€˜((𝐴 Β·β„Ž 𝐡) +β„Ž 𝐢)))
16 fveq2 6892 . . . . . . . 8 (𝑧 = 𝐢 β†’ (π‘‡β€˜π‘§) = (π‘‡β€˜πΆ))
1716oveq2d 7425 . . . . . . 7 (𝑧 = 𝐢 β†’ ((𝐴 Β· (π‘‡β€˜π΅)) + (π‘‡β€˜π‘§)) = ((𝐴 Β· (π‘‡β€˜π΅)) + (π‘‡β€˜πΆ)))
1815, 17eqeq12d 2749 . . . . . 6 (𝑧 = 𝐢 β†’ ((π‘‡β€˜((𝐴 Β·β„Ž 𝐡) +β„Ž 𝑧)) = ((𝐴 Β· (π‘‡β€˜π΅)) + (π‘‡β€˜π‘§)) ↔ (π‘‡β€˜((𝐴 Β·β„Ž 𝐡) +β„Ž 𝐢)) = ((𝐴 Β· (π‘‡β€˜π΅)) + (π‘‡β€˜πΆ))))
197, 13, 18rspc3v 3628 . . . . 5 ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ β„‹ βˆ€π‘§ ∈ β„‹ (π‘‡β€˜((π‘₯ Β·β„Ž 𝑦) +β„Ž 𝑧)) = ((π‘₯ Β· (π‘‡β€˜π‘¦)) + (π‘‡β€˜π‘§)) β†’ (π‘‡β€˜((𝐴 Β·β„Ž 𝐡) +β„Ž 𝐢)) = ((𝐴 Β· (π‘‡β€˜π΅)) + (π‘‡β€˜πΆ))))
202, 19syl5 34 . . . 4 ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝑇 ∈ LinFn β†’ (π‘‡β€˜((𝐴 Β·β„Ž 𝐡) +β„Ž 𝐢)) = ((𝐴 Β· (π‘‡β€˜π΅)) + (π‘‡β€˜πΆ))))
21203expb 1121 . . 3 ((𝐴 ∈ β„‚ ∧ (𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹)) β†’ (𝑇 ∈ LinFn β†’ (π‘‡β€˜((𝐴 Β·β„Ž 𝐡) +β„Ž 𝐢)) = ((𝐴 Β· (π‘‡β€˜π΅)) + (π‘‡β€˜πΆ))))
2221impcom 409 . 2 ((𝑇 ∈ LinFn ∧ (𝐴 ∈ β„‚ ∧ (𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹))) β†’ (π‘‡β€˜((𝐴 Β·β„Ž 𝐡) +β„Ž 𝐢)) = ((𝐴 Β· (π‘‡β€˜π΅)) + (π‘‡β€˜πΆ)))
2322anassrs 469 1 (((𝑇 ∈ LinFn ∧ 𝐴 ∈ β„‚) ∧ (𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹)) β†’ (π‘‡β€˜((𝐴 Β·β„Ž 𝐡) +β„Ž 𝐢)) = ((𝐴 Β· (π‘‡β€˜π΅)) + (π‘‡β€˜πΆ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  β„‚cc 11108   + caddc 11113   Β· cmul 11115   β„‹chba 30172   +β„Ž cva 30173   Β·β„Ž csm 30174  LinFnclf 30207
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-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-hilex 30252
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-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-lnfn 31101
This theorem is referenced by:  lnfnli  31293
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