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Theorem strlem2 30034
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 6645 . . . . 5 (𝑥 = 𝐶 → (proj𝑥) = (proj𝐶))
21fveq1d 6647 . . . 4 (𝑥 = 𝐶 → ((proj𝑥)‘𝑢) = ((proj𝐶)‘𝑢))
32fveq2d 6649 . . 3 (𝑥 = 𝐶 → (norm‘((proj𝑥)‘𝑢)) = (norm‘((proj𝐶)‘𝑢)))
43oveq1d 7150 . 2 (𝑥 = 𝐶 → ((norm‘((proj𝑥)‘𝑢))↑2) = ((norm‘((proj𝐶)‘𝑢))↑2))
5 strlem2.1 . 2 𝑆 = (𝑥C ↦ ((norm‘((proj𝑥)‘𝑢))↑2))
6 ovex 7168 . 2 ((norm‘((proj𝐶)‘𝑢))↑2) ∈ V
74, 5, 6fvmpt 6745 1 (𝐶C → (𝑆𝐶) = ((norm‘((proj𝐶)‘𝑢))↑2))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2111  cmpt 5110  cfv 6324  (class class class)co 7135  2c2 11680  cexp 13425  normcno 28706   C cch 28712  projcpjh 28720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138
This theorem is referenced by:  strlem3a  30035  strlem4  30037  strlem5  30038  jplem2  30052
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