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Theorem strlem2 29955
Description: Lemma for strong state theorem. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)
Hypothesis
Ref Expression
strlem2.1 𝑆 = (𝑥C ↦ ((norm‘((proj𝑥)‘𝑢))↑2))
Assertion
Ref Expression
strlem2 (𝐶C → (𝑆𝐶) = ((norm‘((proj𝐶)‘𝑢))↑2))
Distinct variable groups:   𝑥,𝐶   𝑥,𝑢
Allowed substitution hints:   𝐶(𝑢)   𝑆(𝑥,𝑢)

Proof of Theorem strlem2
StepHypRef Expression
1 fveq2 6663 . . . . 5 (𝑥 = 𝐶 → (proj𝑥) = (proj𝐶))
21fveq1d 6665 . . . 4 (𝑥 = 𝐶 → ((proj𝑥)‘𝑢) = ((proj𝐶)‘𝑢))
32fveq2d 6667 . . 3 (𝑥 = 𝐶 → (norm‘((proj𝑥)‘𝑢)) = (norm‘((proj𝐶)‘𝑢)))
43oveq1d 7160 . 2 (𝑥 = 𝐶 → ((norm‘((proj𝑥)‘𝑢))↑2) = ((norm‘((proj𝐶)‘𝑢))↑2))
5 strlem2.1 . 2 𝑆 = (𝑥C ↦ ((norm‘((proj𝑥)‘𝑢))↑2))
6 ovex 7178 . 2 ((norm‘((proj𝐶)‘𝑢))↑2) ∈ V
74, 5, 6fvmpt 6761 1 (𝐶C → (𝑆𝐶) = ((norm‘((proj𝐶)‘𝑢))↑2))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  cmpt 5137  cfv 6348  (class class class)co 7145  2c2 11680  cexp 13417  normcno 28627   C cch 28633  projcpjh 28641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148
This theorem is referenced by:  strlem3a  29956  strlem4  29958  strlem5  29959  jplem2  29973
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