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Theorem paddfval 39780
Description: Projective subspace sum operation. (Contributed by NM, 29-Dec-2011.)
Hypotheses
Ref Expression
paddfval.l = (le‘𝐾)
paddfval.j = (join‘𝐾)
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 3499 . 2 (𝐾𝐵𝐾 ∈ V)
2 paddfval.p . . 3 + = (+𝑃𝐾)
3 fveq2 6907 . . . . . . 7 ( = 𝐾 → (Atoms‘) = (Atoms‘𝐾))
4 paddfval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4eqtr4di 2793 . . . . . 6 ( = 𝐾 → (Atoms‘) = 𝐴)
65pweqd 4622 . . . . 5 ( = 𝐾 → 𝒫 (Atoms‘) = 𝒫 𝐴)
7 eqidd 2736 . . . . . . . . 9 ( = 𝐾𝑝 = 𝑝)
8 fveq2 6907 . . . . . . . . . 10 ( = 𝐾 → (le‘) = (le‘𝐾))
9 paddfval.l . . . . . . . . . 10 = (le‘𝐾)
108, 9eqtr4di 2793 . . . . . . . . 9 ( = 𝐾 → (le‘) = )
11 fveq2 6907 . . . . . . . . . . 11 ( = 𝐾 → (join‘) = (join‘𝐾))
12 paddfval.j . . . . . . . . . . 11 = (join‘𝐾)
1311, 12eqtr4di 2793 . . . . . . . . . 10 ( = 𝐾 → (join‘) = )
1413oveqd 7448 . . . . . . . . 9 ( = 𝐾 → (𝑞(join‘)𝑟) = (𝑞 𝑟))
157, 10, 14breq123d 5162 . . . . . . . 8 ( = 𝐾 → (𝑝(le‘)(𝑞(join‘)𝑟) ↔ 𝑝 (𝑞 𝑟)))
16152rexbidv 3220 . . . . . . 7 ( = 𝐾 → (∃𝑞𝑚𝑟𝑛 𝑝(le‘)(𝑞(join‘)𝑟) ↔ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)))
175, 16rabeqbidv 3452 . . . . . 6 ( = 𝐾 → {𝑝 ∈ (Atoms‘) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘)(𝑞(join‘)𝑟)} = {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)})
1817uneq2d 4178 . . . . 5 ( = 𝐾 → ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘)(𝑞(join‘)𝑟)}) = ((𝑚𝑛) ∪ {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)}))
196, 6, 18mpoeq123dv 7508 . . . 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 39779 . . . 4 +𝑃 = ( ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘), 𝑛 ∈ 𝒫 (Atoms‘) ↦ ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘)(𝑞(join‘)𝑟)})))
214fvexi 6921 . . . . . 6 𝐴 ∈ V
2221pwex 5386 . . . . 5 𝒫 𝐴 ∈ V
2322, 22mpoex 8103 . . . 4 (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚𝑛) ∪ {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)})) ∈ V
2419, 20, 23fvmpt 7016 . . 3 (𝐾 ∈ V → (+𝑃𝐾) = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚𝑛) ∪ {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)})))
252, 24eqtrid 2787 . 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 1537  wcel 2106  wrex 3068  {crab 3433  Vcvv 3478  cun 3961  𝒫 cpw 4605   class class class wbr 5148  cfv 6563  (class class class)co 7431  cmpo 7433  lecple 17305  joincjn 18369  Atomscatm 39245  +𝑃cpadd 39778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-padd 39779
This theorem is referenced by:  paddval  39781
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