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Theorem lnosub 29743
Description: Subtraction property of a linear operator. (Contributed by NM, 7-Dec-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnosub.1 𝑋 = (BaseSetβ€˜π‘ˆ)
lnosub.5 𝑀 = ( βˆ’π‘£ β€˜π‘ˆ)
lnosub.6 𝑁 = ( βˆ’π‘£ β€˜π‘Š)
lnosub.7 𝐿 = (π‘ˆ LnOp π‘Š)
Assertion
Ref Expression
lnosub (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (π‘‡β€˜(𝐴𝑀𝐡)) = ((π‘‡β€˜π΄)𝑁(π‘‡β€˜π΅)))

Proof of Theorem lnosub
StepHypRef Expression
1 neg1cn 12272 . . . 4 -1 ∈ β„‚
2 lnosub.1 . . . . 5 𝑋 = (BaseSetβ€˜π‘ˆ)
3 eqid 2733 . . . . 5 (BaseSetβ€˜π‘Š) = (BaseSetβ€˜π‘Š)
4 eqid 2733 . . . . 5 ( +𝑣 β€˜π‘ˆ) = ( +𝑣 β€˜π‘ˆ)
5 eqid 2733 . . . . 5 ( +𝑣 β€˜π‘Š) = ( +𝑣 β€˜π‘Š)
6 eqid 2733 . . . . 5 ( ·𝑠OLD β€˜π‘ˆ) = ( ·𝑠OLD β€˜π‘ˆ)
7 eqid 2733 . . . . 5 ( ·𝑠OLD β€˜π‘Š) = ( ·𝑠OLD β€˜π‘Š)
8 lnosub.7 . . . . 5 𝐿 = (π‘ˆ LnOp π‘Š)
92, 3, 4, 5, 6, 7, 8lnolin 29738 . . . 4 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (-1 ∈ β„‚ ∧ 𝐡 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) β†’ (π‘‡β€˜((-1( ·𝑠OLD β€˜π‘ˆ)𝐡)( +𝑣 β€˜π‘ˆ)𝐴)) = ((-1( ·𝑠OLD β€˜π‘Š)(π‘‡β€˜π΅))( +𝑣 β€˜π‘Š)(π‘‡β€˜π΄)))
101, 9mp3anr1 1459 . . 3 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐡 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) β†’ (π‘‡β€˜((-1( ·𝑠OLD β€˜π‘ˆ)𝐡)( +𝑣 β€˜π‘ˆ)𝐴)) = ((-1( ·𝑠OLD β€˜π‘Š)(π‘‡β€˜π΅))( +𝑣 β€˜π‘Š)(π‘‡β€˜π΄)))
1110ancom2s 649 . 2 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (π‘‡β€˜((-1( ·𝑠OLD β€˜π‘ˆ)𝐡)( +𝑣 β€˜π‘ˆ)𝐴)) = ((-1( ·𝑠OLD β€˜π‘Š)(π‘‡β€˜π΅))( +𝑣 β€˜π‘Š)(π‘‡β€˜π΄)))
12 lnosub.5 . . . . . 6 𝑀 = ( βˆ’π‘£ β€˜π‘ˆ)
132, 4, 6, 12nvmval2 29627 . . . . 5 ((π‘ˆ ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝑀𝐡) = ((-1( ·𝑠OLD β€˜π‘ˆ)𝐡)( +𝑣 β€˜π‘ˆ)𝐴))
14133expb 1121 . . . 4 ((π‘ˆ ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝑀𝐡) = ((-1( ·𝑠OLD β€˜π‘ˆ)𝐡)( +𝑣 β€˜π‘ˆ)𝐴))
15143ad2antl1 1186 . . 3 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝑀𝐡) = ((-1( ·𝑠OLD β€˜π‘ˆ)𝐡)( +𝑣 β€˜π‘ˆ)𝐴))
1615fveq2d 6847 . 2 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (π‘‡β€˜(𝐴𝑀𝐡)) = (π‘‡β€˜((-1( ·𝑠OLD β€˜π‘ˆ)𝐡)( +𝑣 β€˜π‘ˆ)𝐴)))
17 simpl2 1193 . . 3 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ π‘Š ∈ NrmCVec)
182, 3, 8lnof 29739 . . . 4 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) β†’ 𝑇:π‘‹βŸΆ(BaseSetβ€˜π‘Š))
19 simpl 484 . . . 4 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
20 ffvelcdm 7033 . . . 4 ((𝑇:π‘‹βŸΆ(BaseSetβ€˜π‘Š) ∧ 𝐴 ∈ 𝑋) β†’ (π‘‡β€˜π΄) ∈ (BaseSetβ€˜π‘Š))
2118, 19, 20syl2an 597 . . 3 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (π‘‡β€˜π΄) ∈ (BaseSetβ€˜π‘Š))
22 simpr 486 . . . 4 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ 𝐡 ∈ 𝑋)
23 ffvelcdm 7033 . . . 4 ((𝑇:π‘‹βŸΆ(BaseSetβ€˜π‘Š) ∧ 𝐡 ∈ 𝑋) β†’ (π‘‡β€˜π΅) ∈ (BaseSetβ€˜π‘Š))
2418, 22, 23syl2an 597 . . 3 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (π‘‡β€˜π΅) ∈ (BaseSetβ€˜π‘Š))
25 lnosub.6 . . . 4 𝑁 = ( βˆ’π‘£ β€˜π‘Š)
263, 5, 7, 25nvmval2 29627 . . 3 ((π‘Š ∈ NrmCVec ∧ (π‘‡β€˜π΄) ∈ (BaseSetβ€˜π‘Š) ∧ (π‘‡β€˜π΅) ∈ (BaseSetβ€˜π‘Š)) β†’ ((π‘‡β€˜π΄)𝑁(π‘‡β€˜π΅)) = ((-1( ·𝑠OLD β€˜π‘Š)(π‘‡β€˜π΅))( +𝑣 β€˜π‘Š)(π‘‡β€˜π΄)))
2717, 21, 24, 26syl3anc 1372 . 2 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((π‘‡β€˜π΄)𝑁(π‘‡β€˜π΅)) = ((-1( ·𝑠OLD β€˜π‘Š)(π‘‡β€˜π΅))( +𝑣 β€˜π‘Š)(π‘‡β€˜π΄)))
2811, 16, 273eqtr4d 2783 1 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (π‘‡β€˜(𝐴𝑀𝐡)) = ((π‘‡β€˜π΄)𝑁(π‘‡β€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  β„‚cc 11054  1c1 11057  -cneg 11391  NrmCVeccnv 29568   +𝑣 cpv 29569  BaseSetcba 29570   ·𝑠OLD cns 29571   βˆ’π‘£ cnsb 29573   LnOp clno 29724
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-po 5546  df-so 5547  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-ltxr 11199  df-sub 11392  df-neg 11393  df-grpo 29477  df-gid 29478  df-ginv 29479  df-gdiv 29480  df-ablo 29529  df-vc 29543  df-nv 29576  df-va 29579  df-ba 29580  df-sm 29581  df-0v 29582  df-vs 29583  df-nmcv 29584  df-lno 29728
This theorem is referenced by:  blometi  29787  blocnilem  29788  ubthlem2  29855
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