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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > istrnN | Structured version Visualization version GIF version |
Description: The predicate "is a translation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
trnset.a | ⊢ 𝐴 = (Atoms‘𝐾) |
trnset.s | ⊢ 𝑆 = (PSubSp‘𝐾) |
trnset.p | ⊢ + = (+𝑃‘𝐾) |
trnset.o | ⊢ ⊥ = (⊥𝑃‘𝐾) |
trnset.w | ⊢ 𝑊 = (WAtoms‘𝐾) |
trnset.m | ⊢ 𝑀 = (PAut‘𝐾) |
trnset.l | ⊢ 𝐿 = (Dil‘𝐾) |
trnset.t | ⊢ 𝑇 = (Trn‘𝐾) |
Ref | Expression |
---|---|
istrnN | ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝐹 ∈ (𝑇‘𝐷) ↔ (𝐹 ∈ (𝐿‘𝐷) ∧ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝐹‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝐹‘𝑟)) ∩ ( ⊥ ‘{𝐷}))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trnset.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
2 | trnset.s | . . . 4 ⊢ 𝑆 = (PSubSp‘𝐾) | |
3 | trnset.p | . . . 4 ⊢ + = (+𝑃‘𝐾) | |
4 | trnset.o | . . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
5 | trnset.w | . . . 4 ⊢ 𝑊 = (WAtoms‘𝐾) | |
6 | trnset.m | . . . 4 ⊢ 𝑀 = (PAut‘𝐾) | |
7 | trnset.l | . . . 4 ⊢ 𝐿 = (Dil‘𝐾) | |
8 | trnset.t | . . . 4 ⊢ 𝑇 = (Trn‘𝐾) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | trnsetN 36177 | . . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝑇‘𝐷) = {𝑓 ∈ (𝐿‘𝐷) ∣ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷}))}) |
10 | 9 | eleq2d 2864 | . 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝐹 ∈ (𝑇‘𝐷) ↔ 𝐹 ∈ {𝑓 ∈ (𝐿‘𝐷) ∣ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷}))})) |
11 | fveq1 6410 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑞) = (𝐹‘𝑞)) | |
12 | 11 | oveq2d 6894 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑞 + (𝑓‘𝑞)) = (𝑞 + (𝐹‘𝑞))) |
13 | 12 | ineq1d 4011 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑞 + (𝐹‘𝑞)) ∩ ( ⊥ ‘{𝐷}))) |
14 | fveq1 6410 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑟) = (𝐹‘𝑟)) | |
15 | 14 | oveq2d 6894 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑟 + (𝑓‘𝑟)) = (𝑟 + (𝐹‘𝑟))) |
16 | 15 | ineq1d 4011 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝐹‘𝑟)) ∩ ( ⊥ ‘{𝐷}))) |
17 | 13, 16 | eqeq12d 2814 | . . . 4 ⊢ (𝑓 = 𝐹 → (((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷})) ↔ ((𝑞 + (𝐹‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝐹‘𝑟)) ∩ ( ⊥ ‘{𝐷})))) |
18 | 17 | 2ralbidv 3170 | . . 3 ⊢ (𝑓 = 𝐹 → (∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷})) ↔ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝐹‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝐹‘𝑟)) ∩ ( ⊥ ‘{𝐷})))) |
19 | 18 | elrab 3556 | . 2 ⊢ (𝐹 ∈ {𝑓 ∈ (𝐿‘𝐷) ∣ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷}))} ↔ (𝐹 ∈ (𝐿‘𝐷) ∧ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝐹‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝐹‘𝑟)) ∩ ( ⊥ ‘{𝐷})))) |
20 | 10, 19 | syl6bb 279 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝐹 ∈ (𝑇‘𝐷) ↔ (𝐹 ∈ (𝐿‘𝐷) ∧ ∀𝑞 ∈ (𝑊‘𝐷)∀𝑟 ∈ (𝑊‘𝐷)((𝑞 + (𝐹‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝐹‘𝑟)) ∩ ( ⊥ ‘{𝐷}))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∀wral 3089 {crab 3093 ∩ cin 3768 {csn 4368 ‘cfv 6101 (class class class)co 6878 Atomscatm 35284 PSubSpcpsubsp 35517 +𝑃cpadd 35816 ⊥𝑃cpolN 35923 WAtomscwpointsN 36007 PAutcpautN 36008 DilcdilN 36123 TrnctrnN 36124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-trnN 36128 |
This theorem is referenced by: (None) |
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