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Theorem strlem2 28494
 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 6103 . . . . 5 (𝑥 = 𝐶 → (proj𝑥) = (proj𝐶))
21fveq1d 6105 . . . 4 (𝑥 = 𝐶 → ((proj𝑥)‘𝑢) = ((proj𝐶)‘𝑢))
32fveq2d 6107 . . 3 (𝑥 = 𝐶 → (norm‘((proj𝑥)‘𝑢)) = (norm‘((proj𝐶)‘𝑢)))
43oveq1d 6564 . 2 (𝑥 = 𝐶 → ((norm‘((proj𝑥)‘𝑢))↑2) = ((norm‘((proj𝐶)‘𝑢))↑2))
5 strlem2.1 . 2 𝑆 = (𝑥C ↦ ((norm‘((proj𝑥)‘𝑢))↑2))
6 ovex 6577 . 2 ((norm‘((proj𝐶)‘𝑢))↑2) ∈ V
74, 5, 6fvmpt 6191 1 (𝐶C → (𝑆𝐶) = ((norm‘((proj𝐶)‘𝑢))↑2))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  2c2 10947  ↑cexp 12722  normℎcno 27164   Cℋ cch 27170  projℎcpjh 27178 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552 This theorem is referenced by:  strlem3a  28495  strlem4  28497  strlem5  28498  jplem2  28512
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