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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 37452 | . . 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 2875 | . 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 6644 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑞) = (𝐹‘𝑞)) | |
12 | 11 | oveq2d 7151 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑞 + (𝑓‘𝑞)) = (𝑞 + (𝐹‘𝑞))) |
13 | 12 | ineq1d 4138 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑞 + (𝐹‘𝑞)) ∩ ( ⊥ ‘{𝐷}))) |
14 | fveq1 6644 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑟) = (𝐹‘𝑟)) | |
15 | 14 | oveq2d 7151 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑟 + (𝑓‘𝑟)) = (𝑟 + (𝐹‘𝑟))) |
16 | 15 | ineq1d 4138 | . . . . 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 3164 | . . 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 3628 | . 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 290 | 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 {crab 3110 ∩ cin 3880 {csn 4525 ‘cfv 6324 (class class class)co 7135 Atomscatm 36559 PSubSpcpsubsp 36792 +𝑃cpadd 37091 ⊥𝑃cpolN 37198 WAtomscwpointsN 37282 PAutcpautN 37283 DilcdilN 37398 TrnctrnN 37399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-trnN 37403 |
This theorem is referenced by: (None) |
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