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Theorem strlem2 30901
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 6825 . . . . 5 (𝑥 = 𝐶 → (proj𝑥) = (proj𝐶))
21fveq1d 6827 . . . 4 (𝑥 = 𝐶 → ((proj𝑥)‘𝑢) = ((proj𝐶)‘𝑢))
32fveq2d 6829 . . 3 (𝑥 = 𝐶 → (norm‘((proj𝑥)‘𝑢)) = (norm‘((proj𝐶)‘𝑢)))
43oveq1d 7352 . 2 (𝑥 = 𝐶 → ((norm‘((proj𝑥)‘𝑢))↑2) = ((norm‘((proj𝐶)‘𝑢))↑2))
5 strlem2.1 . 2 𝑆 = (𝑥C ↦ ((norm‘((proj𝑥)‘𝑢))↑2))
6 ovex 7370 . 2 ((norm‘((proj𝐶)‘𝑢))↑2) ∈ V
74, 5, 6fvmpt 6931 1 (𝐶C → (𝑆𝐶) = ((norm‘((proj𝐶)‘𝑢))↑2))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  cmpt 5175  cfv 6479  (class class class)co 7337  2c2 12129  cexp 13883  normcno 29573   C cch 29579  projcpjh 29587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6431  df-fun 6481  df-fv 6487  df-ov 7340
This theorem is referenced by:  strlem3a  30902  strlem4  30904  strlem5  30905  jplem2  30919
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