Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  paddfval Structured version   Visualization version   GIF version

 Description: Projective subspace sum operation. (Contributed by NM, 29-Dec-2011.)
Hypotheses
Ref Expression
paddfval.a 𝐴 = (Atoms‘𝐾)
paddfval.p + = (+𝑃𝐾)
Assertion
Ref Expression
paddfval (𝐾𝐵+ = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚𝑛) ∪ {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)})))
Distinct variable groups:   𝑚,𝑛,𝑝,𝐴   𝑚,𝑞,𝑟,𝐾,𝑛,𝑝
Allowed substitution hints:   𝐴(𝑟,𝑞)   𝐵(𝑚,𝑛,𝑟,𝑞,𝑝)   + (𝑚,𝑛,𝑟,𝑞,𝑝)   (𝑚,𝑛,𝑟,𝑞,𝑝)   (𝑚,𝑛,𝑟,𝑞,𝑝)

Proof of Theorem paddfval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 3429 . 2 (𝐾𝐵𝐾 ∈ V)
2 paddfval.p . . 3 + = (+𝑃𝐾)
3 fveq2 6664 . . . . . . 7 ( = 𝐾 → (Atoms‘) = (Atoms‘𝐾))
4 paddfval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4eqtr4di 2812 . . . . . 6 ( = 𝐾 → (Atoms‘) = 𝐴)
65pweqd 4517 . . . . 5 ( = 𝐾 → 𝒫 (Atoms‘) = 𝒫 𝐴)
7 eqidd 2760 . . . . . . . . 9 ( = 𝐾𝑝 = 𝑝)
8 fveq2 6664 . . . . . . . . . 10 ( = 𝐾 → (le‘) = (le‘𝐾))
9 paddfval.l . . . . . . . . . 10 = (le‘𝐾)
108, 9eqtr4di 2812 . . . . . . . . 9 ( = 𝐾 → (le‘) = )
11 fveq2 6664 . . . . . . . . . . 11 ( = 𝐾 → (join‘) = (join‘𝐾))
12 paddfval.j . . . . . . . . . . 11 = (join‘𝐾)
1311, 12eqtr4di 2812 . . . . . . . . . 10 ( = 𝐾 → (join‘) = )
1413oveqd 7174 . . . . . . . . 9 ( = 𝐾 → (𝑞(join‘)𝑟) = (𝑞 𝑟))
157, 10, 14breq123d 5051 . . . . . . . 8 ( = 𝐾 → (𝑝(le‘)(𝑞(join‘)𝑟) ↔ 𝑝 (𝑞 𝑟)))
16152rexbidv 3225 . . . . . . 7 ( = 𝐾 → (∃𝑞𝑚𝑟𝑛 𝑝(le‘)(𝑞(join‘)𝑟) ↔ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)))
175, 16rabeqbidv 3399 . . . . . 6 ( = 𝐾 → {𝑝 ∈ (Atoms‘) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘)(𝑞(join‘)𝑟)} = {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)})
1817uneq2d 4071 . . . . 5 ( = 𝐾 → ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘)(𝑞(join‘)𝑟)}) = ((𝑚𝑛) ∪ {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)}))
196, 6, 18mpoeq123dv 7230 . . . 4 ( = 𝐾 → (𝑚 ∈ 𝒫 (Atoms‘), 𝑛 ∈ 𝒫 (Atoms‘) ↦ ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘)(𝑞(join‘)𝑟)})) = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚𝑛) ∪ {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)})))
20 df-padd 37408 . . . 4 +𝑃 = ( ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘), 𝑛 ∈ 𝒫 (Atoms‘) ↦ ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘)(𝑞(join‘)𝑟)})))
214fvexi 6678 . . . . . 6 𝐴 ∈ V
2221pwex 5254 . . . . 5 𝒫 𝐴 ∈ V
2322, 22mpoex 7789 . . . 4 (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚𝑛) ∪ {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)})) ∈ V
2419, 20, 23fvmpt 6765 . . 3 (𝐾 ∈ V → (+𝑃𝐾) = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚𝑛) ∪ {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)})))
252, 24syl5eq 2806 . 2 (𝐾 ∈ V → + = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚𝑛) ∪ {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)})))
261, 25syl 17 1 (𝐾𝐵+ = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚𝑛) ∪ {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2112  ∃wrex 3072  {crab 3075  Vcvv 3410   ∪ cun 3859  𝒫 cpw 4498   class class class wbr 5037  ‘cfv 6341  (class class class)co 7157   ∈ cmpo 7159  lecple 16645  joincjn 17635  Atomscatm 36875  +𝑃cpadd 37407 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5161  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-ov 7160  df-oprab 7161  df-mpo 7162  df-1st 7700  df-2nd 7701  df-padd 37408 This theorem is referenced by:  paddval  37410
 Copyright terms: Public domain W3C validator