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Theorem lnomul 30000
Description: Scalar multiplication property of a linear operator. (Contributed by NM, 5-Dec-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnomul.1 𝑋 = (BaseSetβ€˜π‘ˆ)
lnomul.5 𝑅 = ( ·𝑠OLD β€˜π‘ˆ)
lnomul.6 𝑆 = ( ·𝑠OLD β€˜π‘Š)
lnomul.7 𝐿 = (π‘ˆ LnOp π‘Š)
Assertion
Ref Expression
lnomul (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋)) β†’ (π‘‡β€˜(𝐴𝑅𝐡)) = (𝐴𝑆(π‘‡β€˜π΅)))
Proof of Theorem lnomul
StepHypRef Expression
1 simpl 483 . . 3 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋)) β†’ (π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿))
2 simprl 769 . . 3 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋)) β†’ 𝐴 ∈ β„‚)
3 simprr 771 . . 3 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋)) β†’ 𝐡 ∈ 𝑋)
4 simpl1 1191 . . . 4 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋)) β†’ π‘ˆ ∈ NrmCVec)
5 lnomul.1 . . . . 5 𝑋 = (BaseSetβ€˜π‘ˆ)
6 eqid 2732 . . . . 5 (0vecβ€˜π‘ˆ) = (0vecβ€˜π‘ˆ)
75, 6nvzcl 29874 . . . 4 (π‘ˆ ∈ NrmCVec β†’ (0vecβ€˜π‘ˆ) ∈ 𝑋)
84, 7syl 17 . . 3 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋)) β†’ (0vecβ€˜π‘ˆ) ∈ 𝑋)
9 eqid 2732 . . . 4 (BaseSetβ€˜π‘Š) = (BaseSetβ€˜π‘Š)
10 eqid 2732 . . . 4 ( +𝑣 β€˜π‘ˆ) = ( +𝑣 β€˜π‘ˆ)
11 eqid 2732 . . . 4 ( +𝑣 β€˜π‘Š) = ( +𝑣 β€˜π‘Š)
12 lnomul.5 . . . 4 𝑅 = ( ·𝑠OLD β€˜π‘ˆ)
13 lnomul.6 . . . 4 𝑆 = ( ·𝑠OLD β€˜π‘Š)
14 lnomul.7 . . . 4 𝐿 = (π‘ˆ LnOp π‘Š)
155, 9, 10, 11, 12, 13, 14lnolin 29994 . . 3 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋 ∧ (0vecβ€˜π‘ˆ) ∈ 𝑋)) β†’ (π‘‡β€˜((𝐴𝑅𝐡)( +𝑣 β€˜π‘ˆ)(0vecβ€˜π‘ˆ))) = ((𝐴𝑆(π‘‡β€˜π΅))( +𝑣 β€˜π‘Š)(π‘‡β€˜(0vecβ€˜π‘ˆ))))
161, 2, 3, 8, 15syl13anc 1372 . 2 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋)) β†’ (π‘‡β€˜((𝐴𝑅𝐡)( +𝑣 β€˜π‘ˆ)(0vecβ€˜π‘ˆ))) = ((𝐴𝑆(π‘‡β€˜π΅))( +𝑣 β€˜π‘Š)(π‘‡β€˜(0vecβ€˜π‘ˆ))))
175, 12nvscl 29866 . . . . 5 ((π‘ˆ ∈ NrmCVec ∧ 𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝑅𝐡) ∈ 𝑋)
184, 2, 3, 17syl3anc 1371 . . . 4 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝑅𝐡) ∈ 𝑋)
195, 10, 6nv0rid 29875 . . . 4 ((π‘ˆ ∈ NrmCVec ∧ (𝐴𝑅𝐡) ∈ 𝑋) β†’ ((𝐴𝑅𝐡)( +𝑣 β€˜π‘ˆ)(0vecβ€˜π‘ˆ)) = (𝐴𝑅𝐡))
204, 18, 19syl2anc 584 . . 3 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋)) β†’ ((𝐴𝑅𝐡)( +𝑣 β€˜π‘ˆ)(0vecβ€˜π‘ˆ)) = (𝐴𝑅𝐡))
2120fveq2d 6892 . 2 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋)) β†’ (π‘‡β€˜((𝐴𝑅𝐡)( +𝑣 β€˜π‘ˆ)(0vecβ€˜π‘ˆ))) = (π‘‡β€˜(𝐴𝑅𝐡)))
22 eqid 2732 . . . . . 6 (0vecβ€˜π‘Š) = (0vecβ€˜π‘Š)
235, 9, 6, 22, 14lno0 29996 . . . . 5 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) β†’ (π‘‡β€˜(0vecβ€˜π‘ˆ)) = (0vecβ€˜π‘Š))
2423oveq2d 7421 . . . 4 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) β†’ ((𝐴𝑆(π‘‡β€˜π΅))( +𝑣 β€˜π‘Š)(π‘‡β€˜(0vecβ€˜π‘ˆ))) = ((𝐴𝑆(π‘‡β€˜π΅))( +𝑣 β€˜π‘Š)(0vecβ€˜π‘Š)))
2524adantr 481 . . 3 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋)) β†’ ((𝐴𝑆(π‘‡β€˜π΅))( +𝑣 β€˜π‘Š)(π‘‡β€˜(0vecβ€˜π‘ˆ))) = ((𝐴𝑆(π‘‡β€˜π΅))( +𝑣 β€˜π‘Š)(0vecβ€˜π‘Š)))
26 simpl2 1192 . . . 4 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋)) β†’ π‘Š ∈ NrmCVec)
275, 9, 14lnof 29995 . . . . . . 7 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) β†’ 𝑇:π‘‹βŸΆ(BaseSetβ€˜π‘Š))
2827adantr 481 . . . . . 6 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋)) β†’ 𝑇:π‘‹βŸΆ(BaseSetβ€˜π‘Š))
2928, 3ffvelcdmd 7084 . . . . 5 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋)) β†’ (π‘‡β€˜π΅) ∈ (BaseSetβ€˜π‘Š))
309, 13nvscl 29866 . . . . 5 ((π‘Š ∈ NrmCVec ∧ 𝐴 ∈ β„‚ ∧ (π‘‡β€˜π΅) ∈ (BaseSetβ€˜π‘Š)) β†’ (𝐴𝑆(π‘‡β€˜π΅)) ∈ (BaseSetβ€˜π‘Š))
3126, 2, 29, 30syl3anc 1371 . . . 4 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝑆(π‘‡β€˜π΅)) ∈ (BaseSetβ€˜π‘Š))
329, 11, 22nv0rid 29875 . . . 4 ((π‘Š ∈ NrmCVec ∧ (𝐴𝑆(π‘‡β€˜π΅)) ∈ (BaseSetβ€˜π‘Š)) β†’ ((𝐴𝑆(π‘‡β€˜π΅))( +𝑣 β€˜π‘Š)(0vecβ€˜π‘Š)) = (𝐴𝑆(π‘‡β€˜π΅)))
3326, 31, 32syl2anc 584 . . 3 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋)) β†’ ((𝐴𝑆(π‘‡β€˜π΅))( +𝑣 β€˜π‘Š)(0vecβ€˜π‘Š)) = (𝐴𝑆(π‘‡β€˜π΅)))
3425, 33eqtrd 2772 . 2 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋)) β†’ ((𝐴𝑆(π‘‡β€˜π΅))( +𝑣 β€˜π‘Š)(π‘‡β€˜(0vecβ€˜π‘ˆ))) = (𝐴𝑆(π‘‡β€˜π΅)))
3516, 21, 343eqtr3d 2780 1 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ 𝑋)) β†’ (π‘‡β€˜(𝐴𝑅𝐡)) = (𝐴𝑆(π‘‡β€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405  β„‚cc 11104  NrmCVeccnv 29824   +𝑣 cpv 29825  BaseSetcba 29826   ·𝑠OLD cns 29827  0veccn0v 29828   LnOp clno 29980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-po 5587  df-so 5588  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-ltxr 11249  df-sub 11442  df-neg 11443  df-grpo 29733  df-gid 29734  df-ginv 29735  df-ablo 29785  df-vc 29799  df-nv 29832  df-va 29835  df-ba 29836  df-sm 29837  df-0v 29838  df-nmcv 29840  df-lno 29984
This theorem is referenced by:  nmlno0lem  30033  nmblolbii  30039  blocnilem  30044  ubthlem2  30111
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