Theorem strlem2 32835

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 6877	. . . . 5 ⊢ (𝑥 = 𝐶 → (projℎ‘𝑥) = (projℎ‘𝐶))
2	1	fveq1d 6879	. . . 4 ⊢ (𝑥 = 𝐶 → ((projℎ‘𝑥)‘𝑢) = ((projℎ‘𝐶)‘𝑢))
3	2	fveq2d 6881	. . 3 ⊢ (𝑥 = 𝐶 → (normℎ‘((projℎ‘𝑥)‘𝑢)) = (normℎ‘((projℎ‘𝐶)‘𝑢)))
4	3	oveq1d 7427	. 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 7445	. 2 ⊢ ((normℎ‘((projℎ‘𝐶)‘𝑢))↑2) ∈ V
7	4, 5, 6	fvmpt 6985	1 ⊢ (𝐶 ∈ Cℋ → (𝑆‘𝐶) = ((normℎ‘((projℎ‘𝐶)‘𝑢))↑2))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ↦ cmpt 5186  ‘cfv 6531  (class class class)co 7412  2c2 12378  ↑cexp 14184  norm_ℎcno 31507   C_ℋ cch 31513  proj_ℎcpjh 31521

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6487  df-fun 6533  df-fv 6539  df-ov 7415

This theorem is used by:  strlem3a  32836  strlem4  32838  strlem5  32839  jplem2  32853