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Theorem paddfval 34602
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 3202 . 2 (𝐾𝐵𝐾 ∈ V)
2 paddfval.p . . 3 + = (+𝑃𝐾)
3 fveq2 6158 . . . . . . 7 ( = 𝐾 → (Atoms‘) = (Atoms‘𝐾))
4 paddfval.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
53, 4syl6eqr 2673 . . . . . 6 ( = 𝐾 → (Atoms‘) = 𝐴)
65pweqd 4141 . . . . 5 ( = 𝐾 → 𝒫 (Atoms‘) = 𝒫 𝐴)
7 eqidd 2622 . . . . . . . . 9 ( = 𝐾𝑝 = 𝑝)
8 fveq2 6158 . . . . . . . . . 10 ( = 𝐾 → (le‘) = (le‘𝐾))
9 paddfval.l . . . . . . . . . 10 = (le‘𝐾)
108, 9syl6eqr 2673 . . . . . . . . 9 ( = 𝐾 → (le‘) = )
11 fveq2 6158 . . . . . . . . . . 11 ( = 𝐾 → (join‘) = (join‘𝐾))
12 paddfval.j . . . . . . . . . . 11 = (join‘𝐾)
1311, 12syl6eqr 2673 . . . . . . . . . 10 ( = 𝐾 → (join‘) = )
1413oveqd 6632 . . . . . . . . 9 ( = 𝐾 → (𝑞(join‘)𝑟) = (𝑞 𝑟))
157, 10, 14breq123d 4637 . . . . . . . 8 ( = 𝐾 → (𝑝(le‘)(𝑞(join‘)𝑟) ↔ 𝑝 (𝑞 𝑟)))
16152rexbidv 3052 . . . . . . 7 ( = 𝐾 → (∃𝑞𝑚𝑟𝑛 𝑝(le‘)(𝑞(join‘)𝑟) ↔ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)))
175, 16rabeqbidv 3185 . . . . . 6 ( = 𝐾 → {𝑝 ∈ (Atoms‘) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘)(𝑞(join‘)𝑟)} = {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)})
1817uneq2d 3751 . . . . 5 ( = 𝐾 → ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘)(𝑞(join‘)𝑟)}) = ((𝑚𝑛) ∪ {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)}))
196, 6, 18mpt2eq123dv 6682 . . . 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 34601 . . . 4 +𝑃 = ( ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘), 𝑛 ∈ 𝒫 (Atoms‘) ↦ ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘)(𝑞(join‘)𝑟)})))
21 fvex 6168 . . . . . . 7 (Atoms‘𝐾) ∈ V
224, 21eqeltri 2694 . . . . . 6 𝐴 ∈ V
2322pwex 4818 . . . . 5 𝒫 𝐴 ∈ V
2423, 23mpt2ex 7207 . . . 4 (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚𝑛) ∪ {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)})) ∈ V
2519, 20, 24fvmpt 6249 . . 3 (𝐾 ∈ V → (+𝑃𝐾) = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚𝑛) ∪ {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)})))
262, 25syl5eq 2667 . 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 1480  wcel 1987  wrex 2909  {crab 2912  Vcvv 3190  cun 3558  𝒫 cpw 4136   class class class wbr 4623  cfv 5857  (class class class)co 6615  cmpt2 6617  lecple 15888  joincjn 16884  Atomscatm 34069  +𝑃cpadd 34600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-padd 34601
This theorem is referenced by:  paddval  34603
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