Theorem paddfval 36411

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	⊢ + = (+𝑃‘𝐾)

Assertion

Ref	Expression
paddfval	⊢ (𝐾 ∈ 𝐵 → + = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 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 ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3428	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		paddfval.p	. . 3 ⊢ + = (+𝑃‘𝐾)
3		fveq2 6497	. . . . . . 7 ⊢ (ℎ = 𝐾 → (Atoms‘ℎ) = (Atoms‘𝐾))
4		paddfval.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	syl6eqr 2827	. . . . . 6 ⊢ (ℎ = 𝐾 → (Atoms‘ℎ) = 𝐴)
6	5	pweqd 4422	. . . . 5 ⊢ (ℎ = 𝐾 → 𝒫 (Atoms‘ℎ) = 𝒫 𝐴)
7		eqidd 2774	. . . . . . . . 9 ⊢ (ℎ = 𝐾 → 𝑝 = 𝑝)
8		fveq2 6497	. . . . . . . . . 10 ⊢ (ℎ = 𝐾 → (le‘ℎ) = (le‘𝐾))
9		paddfval.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
10	8, 9	syl6eqr 2827	. . . . . . . . 9 ⊢ (ℎ = 𝐾 → (le‘ℎ) = ≤ )
11		fveq2 6497	. . . . . . . . . . 11 ⊢ (ℎ = 𝐾 → (join‘ℎ) = (join‘𝐾))
12		paddfval.j	. . . . . . . . . . 11 ⊢ ∨ = (join‘𝐾)
13	11, 12	syl6eqr 2827	. . . . . . . . . 10 ⊢ (ℎ = 𝐾 → (join‘ℎ) = ∨ )
14	13	oveqd 6992	. . . . . . . . 9 ⊢ (ℎ = 𝐾 → (𝑞(join‘ℎ)𝑟) = (𝑞 ∨ 𝑟))
15	7, 10, 14	breq123d 4940	. . . . . . . 8 ⊢ (ℎ = 𝐾 → (𝑝(le‘ℎ)(𝑞(join‘ℎ)𝑟) ↔ 𝑝 ≤ (𝑞 ∨ 𝑟)))
16	15	2rexbidv 3240	. . . . . . 7 ⊢ (ℎ = 𝐾 → (∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘ℎ)(𝑞(join‘ℎ)𝑟) ↔ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)))
17	5, 16	rabeqbidv 3403	. . . . . 6 ⊢ (ℎ = 𝐾 → {𝑝 ∈ (Atoms‘ℎ) ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘ℎ)(𝑞(join‘ℎ)𝑟)} = {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)})
18	17	uneq2d 4023	. . . . 5 ⊢ (ℎ = 𝐾 → ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ (Atoms‘ℎ) ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘ℎ)(𝑞(join‘ℎ)𝑟)}) = ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)}))
19	6, 6, 18	mpoeq123dv 7046	. . . 4 ⊢ (ℎ = 𝐾 → (𝑚 ∈ 𝒫 (Atoms‘ℎ), 𝑛 ∈ 𝒫 (Atoms‘ℎ) ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ (Atoms‘ℎ) ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘ℎ)(𝑞(join‘ℎ)𝑟)})) = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)})))
20		df-padd 36410	. . . 4 ⊢ +𝑃 = (ℎ ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘ℎ), 𝑛 ∈ 𝒫 (Atoms‘ℎ) ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ (Atoms‘ℎ) ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘ℎ)(𝑞(join‘ℎ)𝑟)})))
21	4	fvexi 6511	. . . . . 6 ⊢ 𝐴 ∈ V
22	21	pwex 5131	. . . . 5 ⊢ 𝒫 𝐴 ∈ V
23	22, 22	mpoex 7584	. . . 4 ⊢ (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)})) ∈ V
24	19, 20, 23	fvmpt 6594	. . 3 ⊢ (𝐾 ∈ V → (+𝑃‘𝐾) = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)})))
25	2, 24	syl5eq 2821	. 2 ⊢ (𝐾 ∈ V → + = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)})))
26	1, 25	syl 17	1 ⊢ (𝐾 ∈ 𝐵 → + = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)})))

Syntax hints:   → wi 4   = wceq 1508   ∈ wcel 2051  ∃wrex 3084  {crab 3087  Vcvv 3410   ∪ cun 3822  𝒫 cpw 4417   class class class wbr 4926  ‘cfv 6186  (class class class)co 6975   ∈ cmpo 6977  lecple 16427  joincjn 17425  Atomscatm 35877  +_𝑃cpadd 36409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-rex 3089  df-reu 3090  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-ov 6978  df-oprab 6979  df-mpo 6980  df-1st 7500  df-2nd 7501  df-padd 36410

This theorem is referenced by:  paddval  36412