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 34461 | . . 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 2673 | . 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 6102 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑞) = (𝐹‘𝑞)) | |
12 | 11 | oveq2d 6565 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑞 + (𝑓‘𝑞)) = (𝑞 + (𝐹‘𝑞))) |
13 | 12 | ineq1d 3775 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑞 + (𝐹‘𝑞)) ∩ ( ⊥ ‘{𝐷}))) |
14 | fveq1 6102 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑟) = (𝐹‘𝑟)) | |
15 | 14 | oveq2d 6565 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑟 + (𝑓‘𝑟)) = (𝑟 + (𝐹‘𝑟))) |
16 | 15 | ineq1d 3775 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝐹‘𝑟)) ∩ ( ⊥ ‘{𝐷}))) |
17 | 13, 16 | eqeq12d 2625 | . . . 4 ⊢ (𝑓 = 𝐹 → (((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷})) ↔ ((𝑞 + (𝐹‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝐹‘𝑟)) ∩ ( ⊥ ‘{𝐷})))) |
18 | 17 | 2ralbidv 2972 | . . 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 3331 | . 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 275 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ∩ cin 3539 {csn 4125 ‘cfv 5804 (class class class)co 6549 Atomscatm 33568 PSubSpcpsubsp 33800 +𝑃cpadd 34099 ⊥𝑃cpolN 34206 WAtomscwpointsN 34290 PAutcpautN 34291 DilcdilN 34406 TrnctrnN 34407 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-trnN 34411 |
This theorem is referenced by: (None) |
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