Theorem strlem2 32081

Description: Lemma for strong state theorem. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)

Hypothesis

Ref	Expression
strlem2.1	⊢ 𝑆 = (𝑥 ∈ Cℋ ↦ ((normℎ‘((projℎ‘𝑥)‘𝑢))↑2))

Assertion

Ref	Expression
strlem2	⊢ (𝐶 ∈ Cℋ → (𝑆‘𝐶) = ((normℎ‘((projℎ‘𝐶)‘𝑢))↑2))

Distinct variable groups:   𝑥,𝐶   𝑥,𝑢

Allowed substitution hints:   𝐶(𝑢)   𝑆(𝑥,𝑢)

Proof of Theorem strlem2

Step	Hyp	Ref	Expression
1		fveq2 6902	. . . . 5 ⊢ (𝑥 = 𝐶 → (projℎ‘𝑥) = (projℎ‘𝐶))
2	1	fveq1d 6904	. . . 4 ⊢ (𝑥 = 𝐶 → ((projℎ‘𝑥)‘𝑢) = ((projℎ‘𝐶)‘𝑢))
3	2	fveq2d 6906	. . 3 ⊢ (𝑥 = 𝐶 → (normℎ‘((projℎ‘𝑥)‘𝑢)) = (normℎ‘((projℎ‘𝐶)‘𝑢)))
4	3	oveq1d 7441	. 2 ⊢ (𝑥 = 𝐶 → ((normℎ‘((projℎ‘𝑥)‘𝑢))↑2) = ((normℎ‘((projℎ‘𝐶)‘𝑢))↑2))
5		strlem2.1	. 2 ⊢ 𝑆 = (𝑥 ∈ Cℋ ↦ ((normℎ‘((projℎ‘𝑥)‘𝑢))↑2))
6		ovex 7459	. 2 ⊢ ((normℎ‘((projℎ‘𝐶)‘𝑢))↑2) ∈ V
7	4, 5, 6	fvmpt 7010	1 ⊢ (𝐶 ∈ Cℋ → (𝑆‘𝐶) = ((normℎ‘((projℎ‘𝐶)‘𝑢))↑2))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098   ↦ cmpt 5235  ‘cfv 6553  (class class class)co 7426  2c2 12305  ↑cexp 14066  norm_ℎcno 30753   C_ℋ cch 30759  proj_ℎcpjh 30767

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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429

This theorem is referenced by:  strlem3a  32082  strlem4  32084  strlem5  32085  jplem2  32099