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Theorem paddfval 38656
Description: Projective subspace sum operation. (Contributed by NM, 29-Dec-2011.)
Hypotheses
Ref Expression
paddfval.l ≀ = (leβ€˜πΎ)
paddfval.j ∨ = (joinβ€˜πΎ)
paddfval.a 𝐴 = (Atomsβ€˜πΎ)
paddfval.p + = (+π‘ƒβ€˜πΎ)
Assertion
Ref Expression
paddfval (𝐾 ∈ 𝐡 β†’ + = (π‘š ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((π‘š βˆͺ 𝑛) βˆͺ {𝑝 ∈ 𝐴 ∣ βˆƒπ‘ž ∈ π‘š βˆƒπ‘Ÿ ∈ 𝑛 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})))
Distinct variable groups:   π‘š,𝑛,𝑝,𝐴   π‘š,π‘ž,π‘Ÿ,𝐾,𝑛,𝑝
Allowed substitution hints:   𝐴(π‘Ÿ,π‘ž)   𝐡(π‘š,𝑛,π‘Ÿ,π‘ž,𝑝)   + (π‘š,𝑛,π‘Ÿ,π‘ž,𝑝)   ∨ (π‘š,𝑛,π‘Ÿ,π‘ž,𝑝)   ≀ (π‘š,𝑛,π‘Ÿ,π‘ž,𝑝)

Proof of Theorem paddfval
Dummy variable β„Ž is distinct from all other variables.
StepHypRef Expression
1 elex 3492 . 2 (𝐾 ∈ 𝐡 β†’ 𝐾 ∈ V)
2 paddfval.p . . 3 + = (+π‘ƒβ€˜πΎ)
3 fveq2 6888 . . . . . . 7 (β„Ž = 𝐾 β†’ (Atomsβ€˜β„Ž) = (Atomsβ€˜πΎ))
4 paddfval.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
53, 4eqtr4di 2790 . . . . . 6 (β„Ž = 𝐾 β†’ (Atomsβ€˜β„Ž) = 𝐴)
65pweqd 4618 . . . . 5 (β„Ž = 𝐾 β†’ 𝒫 (Atomsβ€˜β„Ž) = 𝒫 𝐴)
7 eqidd 2733 . . . . . . . . 9 (β„Ž = 𝐾 β†’ 𝑝 = 𝑝)
8 fveq2 6888 . . . . . . . . . 10 (β„Ž = 𝐾 β†’ (leβ€˜β„Ž) = (leβ€˜πΎ))
9 paddfval.l . . . . . . . . . 10 ≀ = (leβ€˜πΎ)
108, 9eqtr4di 2790 . . . . . . . . 9 (β„Ž = 𝐾 β†’ (leβ€˜β„Ž) = ≀ )
11 fveq2 6888 . . . . . . . . . . 11 (β„Ž = 𝐾 β†’ (joinβ€˜β„Ž) = (joinβ€˜πΎ))
12 paddfval.j . . . . . . . . . . 11 ∨ = (joinβ€˜πΎ)
1311, 12eqtr4di 2790 . . . . . . . . . 10 (β„Ž = 𝐾 β†’ (joinβ€˜β„Ž) = ∨ )
1413oveqd 7422 . . . . . . . . 9 (β„Ž = 𝐾 β†’ (π‘ž(joinβ€˜β„Ž)π‘Ÿ) = (π‘ž ∨ π‘Ÿ))
157, 10, 14breq123d 5161 . . . . . . . 8 (β„Ž = 𝐾 β†’ (𝑝(leβ€˜β„Ž)(π‘ž(joinβ€˜β„Ž)π‘Ÿ) ↔ 𝑝 ≀ (π‘ž ∨ π‘Ÿ)))
16152rexbidv 3219 . . . . . . 7 (β„Ž = 𝐾 β†’ (βˆƒπ‘ž ∈ π‘š βˆƒπ‘Ÿ ∈ 𝑛 𝑝(leβ€˜β„Ž)(π‘ž(joinβ€˜β„Ž)π‘Ÿ) ↔ βˆƒπ‘ž ∈ π‘š βˆƒπ‘Ÿ ∈ 𝑛 𝑝 ≀ (π‘ž ∨ π‘Ÿ)))
175, 16rabeqbidv 3449 . . . . . 6 (β„Ž = 𝐾 β†’ {𝑝 ∈ (Atomsβ€˜β„Ž) ∣ βˆƒπ‘ž ∈ π‘š βˆƒπ‘Ÿ ∈ 𝑛 𝑝(leβ€˜β„Ž)(π‘ž(joinβ€˜β„Ž)π‘Ÿ)} = {𝑝 ∈ 𝐴 ∣ βˆƒπ‘ž ∈ π‘š βˆƒπ‘Ÿ ∈ 𝑛 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})
1817uneq2d 4162 . . . . 5 (β„Ž = 𝐾 β†’ ((π‘š βˆͺ 𝑛) βˆͺ {𝑝 ∈ (Atomsβ€˜β„Ž) ∣ βˆƒπ‘ž ∈ π‘š βˆƒπ‘Ÿ ∈ 𝑛 𝑝(leβ€˜β„Ž)(π‘ž(joinβ€˜β„Ž)π‘Ÿ)}) = ((π‘š βˆͺ 𝑛) βˆͺ {𝑝 ∈ 𝐴 ∣ βˆƒπ‘ž ∈ π‘š βˆƒπ‘Ÿ ∈ 𝑛 𝑝 ≀ (π‘ž ∨ π‘Ÿ)}))
196, 6, 18mpoeq123dv 7480 . . . 4 (β„Ž = 𝐾 β†’ (π‘š ∈ 𝒫 (Atomsβ€˜β„Ž), 𝑛 ∈ 𝒫 (Atomsβ€˜β„Ž) ↦ ((π‘š βˆͺ 𝑛) βˆͺ {𝑝 ∈ (Atomsβ€˜β„Ž) ∣ βˆƒπ‘ž ∈ π‘š βˆƒπ‘Ÿ ∈ 𝑛 𝑝(leβ€˜β„Ž)(π‘ž(joinβ€˜β„Ž)π‘Ÿ)})) = (π‘š ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((π‘š βˆͺ 𝑛) βˆͺ {𝑝 ∈ 𝐴 ∣ βˆƒπ‘ž ∈ π‘š βˆƒπ‘Ÿ ∈ 𝑛 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})))
20 df-padd 38655 . . . 4 +𝑃 = (β„Ž ∈ V ↦ (π‘š ∈ 𝒫 (Atomsβ€˜β„Ž), 𝑛 ∈ 𝒫 (Atomsβ€˜β„Ž) ↦ ((π‘š βˆͺ 𝑛) βˆͺ {𝑝 ∈ (Atomsβ€˜β„Ž) ∣ βˆƒπ‘ž ∈ π‘š βˆƒπ‘Ÿ ∈ 𝑛 𝑝(leβ€˜β„Ž)(π‘ž(joinβ€˜β„Ž)π‘Ÿ)})))
214fvexi 6902 . . . . . 6 𝐴 ∈ V
2221pwex 5377 . . . . 5 𝒫 𝐴 ∈ V
2322, 22mpoex 8062 . . . 4 (π‘š ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((π‘š βˆͺ 𝑛) βˆͺ {𝑝 ∈ 𝐴 ∣ βˆƒπ‘ž ∈ π‘š βˆƒπ‘Ÿ ∈ 𝑛 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})) ∈ V
2419, 20, 23fvmpt 6995 . . 3 (𝐾 ∈ V β†’ (+π‘ƒβ€˜πΎ) = (π‘š ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((π‘š βˆͺ 𝑛) βˆͺ {𝑝 ∈ 𝐴 ∣ βˆƒπ‘ž ∈ π‘š βˆƒπ‘Ÿ ∈ 𝑛 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})))
252, 24eqtrid 2784 . 2 (𝐾 ∈ V β†’ + = (π‘š ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((π‘š βˆͺ 𝑛) βˆͺ {𝑝 ∈ 𝐴 ∣ βˆƒπ‘ž ∈ π‘š βˆƒπ‘Ÿ ∈ 𝑛 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})))
261, 25syl 17 1 (𝐾 ∈ 𝐡 β†’ + = (π‘š ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((π‘š βˆͺ 𝑛) βˆͺ {𝑝 ∈ 𝐴 ∣ βˆƒπ‘ž ∈ π‘š βˆƒπ‘Ÿ ∈ 𝑛 𝑝 ≀ (π‘ž ∨ π‘Ÿ)})))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  {crab 3432  Vcvv 3474   βˆͺ cun 3945  π’« cpw 4601   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407  lecple 17200  joincjn 18260  Atomscatm 38121  +𝑃cpadd 38654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-padd 38655
This theorem is referenced by:  paddval  38657
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