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Mirrors > Home > MPE Home > Th. List > Mathboxes > isdilN | Structured version Visualization version GIF version |
Description: The predicate "is a dilation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dilset.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dilset.s | ⊢ 𝑆 = (PSubSp‘𝐾) |
dilset.w | ⊢ 𝑊 = (WAtoms‘𝐾) |
dilset.m | ⊢ 𝑀 = (PAut‘𝐾) |
dilset.l | ⊢ 𝐿 = (Dil‘𝐾) |
Ref | Expression |
---|---|
isdilN | ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝐹 ∈ (𝐿‘𝐷) ↔ (𝐹 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝐹‘𝑥) = 𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dilset.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
2 | dilset.s | . . . 4 ⊢ 𝑆 = (PSubSp‘𝐾) | |
3 | dilset.w | . . . 4 ⊢ 𝑊 = (WAtoms‘𝐾) | |
4 | dilset.m | . . . 4 ⊢ 𝑀 = (PAut‘𝐾) | |
5 | dilset.l | . . . 4 ⊢ 𝐿 = (Dil‘𝐾) | |
6 | 1, 2, 3, 4, 5 | dilsetN 37281 | . . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝐿‘𝐷) = {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)}) |
7 | 6 | eleq2d 2896 | . 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝐹 ∈ (𝐿‘𝐷) ↔ 𝐹 ∈ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)})) |
8 | fveq1 6662 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥)) | |
9 | 8 | eqeq1d 2821 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) = 𝑥 ↔ (𝐹‘𝑥) = 𝑥)) |
10 | 9 | imbi2d 343 | . . . 4 ⊢ (𝑓 = 𝐹 → ((𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ⊆ (𝑊‘𝐷) → (𝐹‘𝑥) = 𝑥))) |
11 | 10 | ralbidv 3195 | . . 3 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥) ↔ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝐹‘𝑥) = 𝑥))) |
12 | 11 | elrab 3678 | . 2 ⊢ (𝐹 ∈ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)} ↔ (𝐹 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝐹‘𝑥) = 𝑥))) |
13 | 7, 12 | syl6bb 289 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝐹 ∈ (𝐿‘𝐷) ↔ (𝐹 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝐹‘𝑥) = 𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ∀wral 3136 {crab 3140 ⊆ wss 3934 ‘cfv 6348 Atomscatm 36391 PSubSpcpsubsp 36624 WAtomscwpointsN 37114 PAutcpautN 37115 DilcdilN 37230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-dilN 37234 |
This theorem is referenced by: (None) |
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