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Theorem istrnN 38623
Description: The predicate "is a translation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
trnset.a 𝐴 = (Atomsβ€˜πΎ)
trnset.s 𝑆 = (PSubSpβ€˜πΎ)
trnset.p + = (+π‘ƒβ€˜πΎ)
trnset.o βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
trnset.w π‘Š = (WAtomsβ€˜πΎ)
trnset.m 𝑀 = (PAutβ€˜πΎ)
trnset.l 𝐿 = (Dilβ€˜πΎ)
trnset.t 𝑇 = (Trnβ€˜πΎ)
Assertion
Ref Expression
istrnN ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (𝐹 ∈ (π‘‡β€˜π·) ↔ (𝐹 ∈ (πΏβ€˜π·) ∧ βˆ€π‘ž ∈ (π‘Šβ€˜π·)βˆ€π‘Ÿ ∈ (π‘Šβ€˜π·)((π‘ž + (πΉβ€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (πΉβ€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷})))))
Distinct variable groups:   π‘Ÿ,π‘ž,𝐾   π‘Š,π‘ž,π‘Ÿ   𝐷,π‘ž,π‘Ÿ   𝐹,π‘ž,π‘Ÿ
Allowed substitution hints:   𝐴(π‘Ÿ,π‘ž)   𝐡(π‘Ÿ,π‘ž)   + (π‘Ÿ,π‘ž)   𝑆(π‘Ÿ,π‘ž)   𝑇(π‘Ÿ,π‘ž)   𝐿(π‘Ÿ,π‘ž)   𝑀(π‘Ÿ,π‘ž)   βŠ₯ (π‘Ÿ,π‘ž)

Proof of Theorem istrnN
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 trnset.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
2 trnset.s . . . 4 𝑆 = (PSubSpβ€˜πΎ)
3 trnset.p . . . 4 + = (+π‘ƒβ€˜πΎ)
4 trnset.o . . . 4 βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)
5 trnset.w . . . 4 π‘Š = (WAtomsβ€˜πΎ)
6 trnset.m . . . 4 𝑀 = (PAutβ€˜πΎ)
7 trnset.l . . . 4 𝐿 = (Dilβ€˜πΎ)
8 trnset.t . . . 4 𝑇 = (Trnβ€˜πΎ)
91, 2, 3, 4, 5, 6, 7, 8trnsetN 38622 . . 3 ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (π‘‡β€˜π·) = {𝑓 ∈ (πΏβ€˜π·) ∣ βˆ€π‘ž ∈ (π‘Šβ€˜π·)βˆ€π‘Ÿ ∈ (π‘Šβ€˜π·)((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷}))})
109eleq2d 2824 . 2 ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (𝐹 ∈ (π‘‡β€˜π·) ↔ 𝐹 ∈ {𝑓 ∈ (πΏβ€˜π·) ∣ βˆ€π‘ž ∈ (π‘Šβ€˜π·)βˆ€π‘Ÿ ∈ (π‘Šβ€˜π·)((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷}))}))
11 fveq1 6842 . . . . . . 7 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘ž) = (πΉβ€˜π‘ž))
1211oveq2d 7374 . . . . . 6 (𝑓 = 𝐹 β†’ (π‘ž + (π‘“β€˜π‘ž)) = (π‘ž + (πΉβ€˜π‘ž)))
1312ineq1d 4172 . . . . 5 (𝑓 = 𝐹 β†’ ((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘ž + (πΉβ€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})))
14 fveq1 6842 . . . . . . 7 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘Ÿ) = (πΉβ€˜π‘Ÿ))
1514oveq2d 7374 . . . . . 6 (𝑓 = 𝐹 β†’ (π‘Ÿ + (π‘“β€˜π‘Ÿ)) = (π‘Ÿ + (πΉβ€˜π‘Ÿ)))
1615ineq1d 4172 . . . . 5 (𝑓 = 𝐹 β†’ ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (πΉβ€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷})))
1713, 16eqeq12d 2753 . . . 4 (𝑓 = 𝐹 β†’ (((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷})) ↔ ((π‘ž + (πΉβ€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (πΉβ€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷}))))
18172ralbidv 3213 . . 3 (𝑓 = 𝐹 β†’ (βˆ€π‘ž ∈ (π‘Šβ€˜π·)βˆ€π‘Ÿ ∈ (π‘Šβ€˜π·)((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷})) ↔ βˆ€π‘ž ∈ (π‘Šβ€˜π·)βˆ€π‘Ÿ ∈ (π‘Šβ€˜π·)((π‘ž + (πΉβ€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (πΉβ€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷}))))
1918elrab 3646 . 2 (𝐹 ∈ {𝑓 ∈ (πΏβ€˜π·) ∣ βˆ€π‘ž ∈ (π‘Šβ€˜π·)βˆ€π‘Ÿ ∈ (π‘Šβ€˜π·)((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷}))} ↔ (𝐹 ∈ (πΏβ€˜π·) ∧ βˆ€π‘ž ∈ (π‘Šβ€˜π·)βˆ€π‘Ÿ ∈ (π‘Šβ€˜π·)((π‘ž + (πΉβ€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (πΉβ€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷}))))
2010, 19bitrdi 287 1 ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (𝐹 ∈ (π‘‡β€˜π·) ↔ (𝐹 ∈ (πΏβ€˜π·) ∧ βˆ€π‘ž ∈ (π‘Šβ€˜π·)βˆ€π‘Ÿ ∈ (π‘Šβ€˜π·)((π‘ž + (πΉβ€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (πΉβ€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷})))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3408   ∩ cin 3910  {csn 4587  β€˜cfv 6497  (class class class)co 7358  Atomscatm 37728  PSubSpcpsubsp 37962  +𝑃cpadd 38261  βŠ₯𝑃cpolN 38368  WAtomscwpointsN 38452  PAutcpautN 38453  DilcdilN 38568  TrnctrnN 38569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-trnN 38573
This theorem is referenced by: (None)
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