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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 39855 | . . 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 2812 | . 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 6900 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑞) = (𝐹‘𝑞)) | |
12 | 11 | oveq2d 7440 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑞 + (𝑓‘𝑞)) = (𝑞 + (𝐹‘𝑞))) |
13 | 12 | ineq1d 4212 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑞 + (𝐹‘𝑞)) ∩ ( ⊥ ‘{𝐷}))) |
14 | fveq1 6900 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑟) = (𝐹‘𝑟)) | |
15 | 14 | oveq2d 7440 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑟 + (𝑓‘𝑟)) = (𝑟 + (𝐹‘𝑟))) |
16 | 15 | ineq1d 4212 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝐹‘𝑟)) ∩ ( ⊥ ‘{𝐷}))) |
17 | 13, 16 | eqeq12d 2742 | . . . 4 ⊢ (𝑓 = 𝐹 → (((𝑞 + (𝑓‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝑓‘𝑟)) ∩ ( ⊥ ‘{𝐷})) ↔ ((𝑞 + (𝐹‘𝑞)) ∩ ( ⊥ ‘{𝐷})) = ((𝑟 + (𝐹‘𝑟)) ∩ ( ⊥ ‘{𝐷})))) |
18 | 17 | 2ralbidv 3209 | . . 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 3681 | . 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 | bitrdi 286 | 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 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 {crab 3419 ∩ cin 3946 {csn 4633 ‘cfv 6554 (class class class)co 7424 Atomscatm 38961 PSubSpcpsubsp 39195 +𝑃cpadd 39494 ⊥𝑃cpolN 39601 WAtomscwpointsN 39685 PAutcpautN 39686 DilcdilN 39801 TrnctrnN 39802 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-trnN 39806 |
This theorem is referenced by: (None) |
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