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Theorem strlem2 32008
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 6884 . . . . 5 (𝑥 = 𝐶 → (proj𝑥) = (proj𝐶))
21fveq1d 6886 . . . 4 (𝑥 = 𝐶 → ((proj𝑥)‘𝑢) = ((proj𝐶)‘𝑢))
32fveq2d 6888 . . 3 (𝑥 = 𝐶 → (norm‘((proj𝑥)‘𝑢)) = (norm‘((proj𝐶)‘𝑢)))
43oveq1d 7419 . 2 (𝑥 = 𝐶 → ((norm‘((proj𝑥)‘𝑢))↑2) = ((norm‘((proj𝐶)‘𝑢))↑2))
5 strlem2.1 . 2 𝑆 = (𝑥C ↦ ((norm‘((proj𝑥)‘𝑢))↑2))
6 ovex 7437 . 2 ((norm‘((proj𝐶)‘𝑢))↑2) ∈ V
74, 5, 6fvmpt 6991 1 (𝐶C → (𝑆𝐶) = ((norm‘((proj𝐶)‘𝑢))↑2))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cmpt 5224  cfv 6536  (class class class)co 7404  2c2 12268  cexp 14029  normcno 30680   C cch 30686  projcpjh 30694
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407
This theorem is referenced by:  strlem3a  32009  strlem4  32011  strlem5  32012  jplem2  32026
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