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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 6825 | . . . . 5 ⊢ (𝑥 = 𝐶 → (projℎ‘𝑥) = (projℎ‘𝐶)) | |
2 | 1 | fveq1d 6827 | . . . 4 ⊢ (𝑥 = 𝐶 → ((projℎ‘𝑥)‘𝑢) = ((projℎ‘𝐶)‘𝑢)) |
3 | 2 | fveq2d 6829 | . . 3 ⊢ (𝑥 = 𝐶 → (normℎ‘((projℎ‘𝑥)‘𝑢)) = (normℎ‘((projℎ‘𝐶)‘𝑢))) |
4 | 3 | oveq1d 7352 | . 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 7370 | . 2 ⊢ ((normℎ‘((projℎ‘𝐶)‘𝑢))↑2) ∈ V | |
7 | 4, 5, 6 | fvmpt 6931 | 1 ⊢ (𝐶 ∈ Cℋ → (𝑆‘𝐶) = ((normℎ‘((projℎ‘𝐶)‘𝑢))↑2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ↦ cmpt 5175 ‘cfv 6479 (class class class)co 7337 2c2 12129 ↑cexp 13883 normℎcno 29573 Cℋ cch 29579 projℎcpjh 29587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pr 5372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-iota 6431 df-fun 6481 df-fv 6487 df-ov 7340 |
This theorem is referenced by: strlem3a 30902 strlem4 30904 strlem5 30905 jplem2 30919 |
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