Theorem paddfval 36932

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 3512	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		paddfval.p	. . 3 ⊢ + = (+𝑃‘𝐾)
3		fveq2 6669	. . . . . . 7 ⊢ (ℎ = 𝐾 → (Atoms‘ℎ) = (Atoms‘𝐾))
4		paddfval.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	syl6eqr 2874	. . . . . 6 ⊢ (ℎ = 𝐾 → (Atoms‘ℎ) = 𝐴)
6	5	pweqd 4557	. . . . 5 ⊢ (ℎ = 𝐾 → 𝒫 (Atoms‘ℎ) = 𝒫 𝐴)
7		eqidd 2822	. . . . . . . . 9 ⊢ (ℎ = 𝐾 → 𝑝 = 𝑝)
8		fveq2 6669	. . . . . . . . . 10 ⊢ (ℎ = 𝐾 → (le‘ℎ) = (le‘𝐾))
9		paddfval.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
10	8, 9	syl6eqr 2874	. . . . . . . . 9 ⊢ (ℎ = 𝐾 → (le‘ℎ) = ≤ )
11		fveq2 6669	. . . . . . . . . . 11 ⊢ (ℎ = 𝐾 → (join‘ℎ) = (join‘𝐾))
12		paddfval.j	. . . . . . . . . . 11 ⊢ ∨ = (join‘𝐾)
13	11, 12	syl6eqr 2874	. . . . . . . . . 10 ⊢ (ℎ = 𝐾 → (join‘ℎ) = ∨ )
14	13	oveqd 7172	. . . . . . . . 9 ⊢ (ℎ = 𝐾 → (𝑞(join‘ℎ)𝑟) = (𝑞 ∨ 𝑟))
15	7, 10, 14	breq123d 5079	. . . . . . . 8 ⊢ (ℎ = 𝐾 → (𝑝(le‘ℎ)(𝑞(join‘ℎ)𝑟) ↔ 𝑝 ≤ (𝑞 ∨ 𝑟)))
16	15	2rexbidv 3300	. . . . . . 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 3485	. . . . . 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 4138	. . . . 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 7228	. . . 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 36931	. . . 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 6683	. . . . . 6 ⊢ 𝐴 ∈ V
22	21	pwex 5280	. . . . 5 ⊢ 𝒫 𝐴 ∈ V
23	22, 22	mpoex 7776	. . . 4 ⊢ (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)})) ∈ V
24	19, 20, 23	fvmpt 6767	. . 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 2868	. 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 1533   ∈ wcel 2110  ∃wrex 3139  {crab 3142  Vcvv 3494   ∪ cun 3933  𝒫 cpw 4538   class class class wbr 5065  ‘cfv 6354  (class class class)co 7155   ∈ cmpo 7157  lecple 16571  joincjn 17553  Atomscatm 36398  +_𝑃cpadd 36930

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689  df-padd 36931

This theorem is referenced by:  paddval  36933