Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pmtridfv1 Structured version   Visualization version   GIF version

Theorem pmtridfv1 31993
Description: Value at X of the transposition of 𝑋 and 𝑌 (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.)
Hypotheses
Ref Expression
pmtridf1o.a (𝜑𝐴𝑉)
pmtridf1o.x (𝜑𝑋𝐴)
pmtridf1o.y (𝜑𝑌𝐴)
pmtridf1o.t 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
Assertion
Ref Expression
pmtridfv1 (𝜑 → (𝑇𝑋) = 𝑌)

Proof of Theorem pmtridfv1
StepHypRef Expression
1 pmtridf1o.t . . . . 5 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
2 simpr 486 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋 = 𝑌)
32iftrued 4495 . . . . 5 ((𝜑𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
41, 3eqtrid 2785 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
54fveq1d 6845 . . 3 ((𝜑𝑋 = 𝑌) → (𝑇𝑋) = (( I ↾ 𝐴)‘𝑋))
6 pmtridf1o.x . . . . 5 (𝜑𝑋𝐴)
7 fvresi 7120 . . . . 5 (𝑋𝐴 → (( I ↾ 𝐴)‘𝑋) = 𝑋)
86, 7syl 17 . . . 4 (𝜑 → (( I ↾ 𝐴)‘𝑋) = 𝑋)
98adantr 482 . . 3 ((𝜑𝑋 = 𝑌) → (( I ↾ 𝐴)‘𝑋) = 𝑋)
105, 9, 23eqtrd 2777 . 2 ((𝜑𝑋 = 𝑌) → (𝑇𝑋) = 𝑌)
11 simpr 486 . . . . . . 7 ((𝜑𝑋𝑌) → 𝑋𝑌)
1211neneqd 2945 . . . . . 6 ((𝜑𝑋𝑌) → ¬ 𝑋 = 𝑌)
1312iffalsed 4498 . . . . 5 ((𝜑𝑋𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
141, 13eqtrid 2785 . . . 4 ((𝜑𝑋𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
1514fveq1d 6845 . . 3 ((𝜑𝑋𝑌) → (𝑇𝑋) = (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑋))
16 pmtridf1o.a . . . . 5 (𝜑𝐴𝑉)
1716adantr 482 . . . 4 ((𝜑𝑋𝑌) → 𝐴𝑉)
186adantr 482 . . . 4 ((𝜑𝑋𝑌) → 𝑋𝐴)
19 pmtridf1o.y . . . . 5 (𝜑𝑌𝐴)
2019adantr 482 . . . 4 ((𝜑𝑋𝑌) → 𝑌𝐴)
21 eqid 2733 . . . . 5 (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
2221pmtrprfv 19240 . . . 4 ((𝐴𝑉 ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑋) = 𝑌)
2317, 18, 20, 11, 22syl13anc 1373 . . 3 ((𝜑𝑋𝑌) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑋) = 𝑌)
2415, 23eqtrd 2773 . 2 ((𝜑𝑋𝑌) → (𝑇𝑋) = 𝑌)
2510, 24pm2.61dane 3029 1 (𝜑 → (𝑇𝑋) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2940  ifcif 4487  {cpr 4589   I cid 5531  cres 5636  cfv 6497  pmTrspcpmtr 19228
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 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  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-pw 4563  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-suc 6324  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-1o 8413  df-2o 8414  df-en 8887  df-pmtr 19229
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator