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Theorem paddfval 34101
 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 3185 . 2 (𝐾𝐵𝐾 ∈ V)
2 paddfval.p . . 3 + = (+𝑃𝐾)
3 fveq2 6103 . . . . . . 7 ( = 𝐾 → (Atoms‘) = (Atoms‘𝐾))
4 paddfval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2662 . . . . . 6 ( = 𝐾 → (Atoms‘) = 𝐴)
65pweqd 4113 . . . . 5 ( = 𝐾 → 𝒫 (Atoms‘) = 𝒫 𝐴)
7 eqidd 2611 . . . . . . . . 9 ( = 𝐾𝑝 = 𝑝)
8 fveq2 6103 . . . . . . . . . 10 ( = 𝐾 → (le‘) = (le‘𝐾))
9 paddfval.l . . . . . . . . . 10 = (le‘𝐾)
108, 9syl6eqr 2662 . . . . . . . . 9 ( = 𝐾 → (le‘) = )
11 fveq2 6103 . . . . . . . . . . 11 ( = 𝐾 → (join‘) = (join‘𝐾))
12 paddfval.j . . . . . . . . . . 11 = (join‘𝐾)
1311, 12syl6eqr 2662 . . . . . . . . . 10 ( = 𝐾 → (join‘) = )
1413oveqd 6566 . . . . . . . . 9 ( = 𝐾 → (𝑞(join‘)𝑟) = (𝑞 𝑟))
157, 10, 14breq123d 4597 . . . . . . . 8 ( = 𝐾 → (𝑝(le‘)(𝑞(join‘)𝑟) ↔ 𝑝 (𝑞 𝑟)))
16152rexbidv 3039 . . . . . . 7 ( = 𝐾 → (∃𝑞𝑚𝑟𝑛 𝑝(le‘)(𝑞(join‘)𝑟) ↔ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)))
175, 16rabeqbidv 3168 . . . . . 6 ( = 𝐾 → {𝑝 ∈ (Atoms‘) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘)(𝑞(join‘)𝑟)} = {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)})
1817uneq2d 3729 . . . . 5 ( = 𝐾 → ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘)(𝑞(join‘)𝑟)}) = ((𝑚𝑛) ∪ {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)}))
196, 6, 18mpt2eq123dv 6615 . . . 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 34100 . . . 4 +𝑃 = ( ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘), 𝑛 ∈ 𝒫 (Atoms‘) ↦ ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘)(𝑞(join‘)𝑟)})))
21 fvex 6113 . . . . . . 7 (Atoms‘𝐾) ∈ V
224, 21eqeltri 2684 . . . . . 6 𝐴 ∈ V
2322pwex 4774 . . . . 5 𝒫 𝐴 ∈ V
2423, 23mpt2ex 7136 . . . 4 (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚𝑛) ∪ {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)})) ∈ V
2519, 20, 24fvmpt 6191 . . 3 (𝐾 ∈ V → (+𝑃𝐾) = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚𝑛) ∪ {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)})))
262, 25syl5eq 2656 . 2 (𝐾 ∈ V → + = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚𝑛) ∪ {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)})))
271, 26syl 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 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173   ∪ cun 3538  𝒫 cpw 4108   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  lecple 15775  joincjn 16767  Atomscatm 33568  +𝑃cpadd 34099 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-padd 34100 This theorem is referenced by:  paddval  34102
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