Theorem strlem2 32640

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 6888	. . . . 5 ⊢ (𝑥 = 𝐶 → (projℎ‘𝑥) = (projℎ‘𝐶))
2	1	fveq1d 6890	. . . 4 ⊢ (𝑥 = 𝐶 → ((projℎ‘𝑥)‘𝑢) = ((projℎ‘𝐶)‘𝑢))
3	2	fveq2d 6892	. . 3 ⊢ (𝑥 = 𝐶 → (normℎ‘((projℎ‘𝑥)‘𝑢)) = (normℎ‘((projℎ‘𝐶)‘𝑢)))
4	3	oveq1d 7438	. 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 7456	. 2 ⊢ ((normℎ‘((projℎ‘𝐶)‘𝑢))↑2) ∈ V
7	4, 5, 6	fvmpt 6996	1 ⊢ (𝐶 ∈ Cℋ → (𝑆‘𝐶) = ((normℎ‘((projℎ‘𝐶)‘𝑢))↑2))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146   ↦ cmpt 5197  ‘cfv 6543  (class class class)co 7423  2c2 12313  ↑cexp 14117  norm_ℎcno 31312   C_ℋ cch 31318  proj_ℎcpjh 31326

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426

This theorem is used by:  strlem3a  32641  strlem4  32643  strlem5  32644  jplem2  32658