Theorem paddfval 41120

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 3475	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		paddfval.p	. . 3 ⊢ + = (+𝑃‘𝐾)
3		fveq2 6916	. . . . . . 7 ⊢ (ℎ = 𝐾 → (Atoms‘ℎ) = (Atoms‘𝐾))
4		paddfval.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2816	. . . . . 6 ⊢ (ℎ = 𝐾 → (Atoms‘ℎ) = 𝐴)
6	5	pweqd 4577	. . . . 5 ⊢ (ℎ = 𝐾 → 𝒫 (Atoms‘ℎ) = 𝒫 𝐴)
7		eqidd 2764	. . . . . . . . 9 ⊢ (ℎ = 𝐾 → 𝑝 = 𝑝)
8		fveq2 6916	. . . . . . . . . 10 ⊢ (ℎ = 𝐾 → (le‘ℎ) = (le‘𝐾))
9		paddfval.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
10	8, 9	eqtr4di 2816	. . . . . . . . 9 ⊢ (ℎ = 𝐾 → (le‘ℎ) = ≤ )
11		fveq2 6916	. . . . . . . . . . 11 ⊢ (ℎ = 𝐾 → (join‘ℎ) = (join‘𝐾))
12		paddfval.j	. . . . . . . . . . 11 ⊢ ∨ = (join‘𝐾)
13	11, 12	eqtr4di 2816	. . . . . . . . . 10 ⊢ (ℎ = 𝐾 → (join‘ℎ) = ∨ )
14	13	oveqd 7468	. . . . . . . . 9 ⊢ (ℎ = 𝐾 → (𝑞(join‘ℎ)𝑟) = (𝑞 ∨ 𝑟))
15	7, 10, 14	breq123d 5122	. . . . . . . 8 ⊢ (ℎ = 𝐾 → (𝑝(le‘ℎ)(𝑞(join‘ℎ)𝑟) ↔ 𝑝 ≤ (𝑞 ∨ 𝑟)))
16	15	2rexbidv 3231	. . . . . . 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 3433	. . . . . 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 4170	. . . . 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 7526	. . . 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 41119	. . . 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 6930	. . . . . 6 ⊢ 𝐴 ∈ V
22	21	pwex 5349	. . . . 5 ⊢ 𝒫 𝐴 ∈ V
23	22, 22	mpoex 8127	. . . 4 ⊢ (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)})) ∈ V
24	19, 20, 23	fvmpt 7024	. . 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	eqtrid 2810	. 2 ⊢ (𝐾 ∈ V → + = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)})))
26	1, 25	syl 18	1 ⊢ (𝐾 ∈ 𝐵 → + = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)})))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  {crab 3416  Vcvv 3454   ∪ cun 4152  𝒫 cpw 4560   class class class wbr 5108  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  lecple 17479  joincjn 18535  Atomscatm 40586  +_𝑃cpadd 41118

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-padd 41119

This theorem is used by:  paddval  41121