Theorem strlem2 32859

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 6886	. . . . 5 ⊢ (𝑥 = 𝐶 → (projℎ‘𝑥) = (projℎ‘𝐶))
2	1	fveq1d 6888	. . . 4 ⊢ (𝑥 = 𝐶 → ((projℎ‘𝑥)‘𝑢) = ((projℎ‘𝐶)‘𝑢))
3	2	fveq2d 6890	. . 3 ⊢ (𝑥 = 𝐶 → (normℎ‘((projℎ‘𝑥)‘𝑢)) = (normℎ‘((projℎ‘𝐶)‘𝑢)))
4	3	oveq1d 7436	. 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 7454	. 2 ⊢ ((normℎ‘((projℎ‘𝐶)‘𝑢))↑2) ∈ V
7	4, 5, 6	fvmpt 6994	1 ⊢ (𝐶 ∈ Cℋ → (𝑆‘𝐶) = ((normℎ‘((projℎ‘𝐶)‘𝑢))↑2))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   ↦ cmpt 5185  ‘cfv 6539  (class class class)co 7421  2c2 12401  ↑cexp 14208  norm_ℎcno 31531   C_ℋ cch 31537  proj_ℎcpjh 31545

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6541  df-fv 6547  df-ov 7424

This theorem is used by:  strlem3a  32860  strlem4  32862  strlem5  32863  jplem2  32877