Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > strlem2 | Structured version Visualization version GIF version |
Description: Lemma for strong state theorem. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
strlem2.1 | ⊢ 𝑆 = (𝑥 ∈ Cℋ ↦ ((normℎ‘((projℎ‘𝑥)‘𝑢))↑2)) |
Ref | Expression |
---|---|
strlem2 | ⊢ (𝐶 ∈ Cℋ → (𝑆‘𝐶) = ((normℎ‘((projℎ‘𝐶)‘𝑢))↑2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6191 | . . . . 5 ⊢ (𝑥 = 𝐶 → (projℎ‘𝑥) = (projℎ‘𝐶)) | |
2 | 1 | fveq1d 6193 | . . . 4 ⊢ (𝑥 = 𝐶 → ((projℎ‘𝑥)‘𝑢) = ((projℎ‘𝐶)‘𝑢)) |
3 | 2 | fveq2d 6195 | . . 3 ⊢ (𝑥 = 𝐶 → (normℎ‘((projℎ‘𝑥)‘𝑢)) = (normℎ‘((projℎ‘𝐶)‘𝑢))) |
4 | 3 | oveq1d 6665 | . 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 6678 | . 2 ⊢ ((normℎ‘((projℎ‘𝐶)‘𝑢))↑2) ∈ V | |
7 | 4, 5, 6 | fvmpt 6282 | 1 ⊢ (𝐶 ∈ Cℋ → (𝑆‘𝐶) = ((normℎ‘((projℎ‘𝐶)‘𝑢))↑2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 2c2 11070 ↑cexp 12860 normℎcno 27780 Cℋ cch 27786 projℎcpjh 27794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 |
This theorem is referenced by: strlem3a 29111 strlem4 29113 strlem5 29114 jplem2 29128 |
Copyright terms: Public domain | W3C validator |