Theorem paddfval 39816

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 3480	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		paddfval.p	. . 3 ⊢ + = (+𝑃‘𝐾)
3		fveq2 6876	. . . . . . 7 ⊢ (ℎ = 𝐾 → (Atoms‘ℎ) = (Atoms‘𝐾))
4		paddfval.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2788	. . . . . 6 ⊢ (ℎ = 𝐾 → (Atoms‘ℎ) = 𝐴)
6	5	pweqd 4592	. . . . 5 ⊢ (ℎ = 𝐾 → 𝒫 (Atoms‘ℎ) = 𝒫 𝐴)
7		eqidd 2736	. . . . . . . . 9 ⊢ (ℎ = 𝐾 → 𝑝 = 𝑝)
8		fveq2 6876	. . . . . . . . . 10 ⊢ (ℎ = 𝐾 → (le‘ℎ) = (le‘𝐾))
9		paddfval.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
10	8, 9	eqtr4di 2788	. . . . . . . . 9 ⊢ (ℎ = 𝐾 → (le‘ℎ) = ≤ )
11		fveq2 6876	. . . . . . . . . . 11 ⊢ (ℎ = 𝐾 → (join‘ℎ) = (join‘𝐾))
12		paddfval.j	. . . . . . . . . . 11 ⊢ ∨ = (join‘𝐾)
13	11, 12	eqtr4di 2788	. . . . . . . . . 10 ⊢ (ℎ = 𝐾 → (join‘ℎ) = ∨ )
14	13	oveqd 7422	. . . . . . . . 9 ⊢ (ℎ = 𝐾 → (𝑞(join‘ℎ)𝑟) = (𝑞 ∨ 𝑟))
15	7, 10, 14	breq123d 5133	. . . . . . . 8 ⊢ (ℎ = 𝐾 → (𝑝(le‘ℎ)(𝑞(join‘ℎ)𝑟) ↔ 𝑝 ≤ (𝑞 ∨ 𝑟)))
16	15	2rexbidv 3206	. . . . . . 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 3434	. . . . . 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 4143	. . . . 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 7482	. . . 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 39815	. . . 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 6890	. . . . . 6 ⊢ 𝐴 ∈ V
22	21	pwex 5350	. . . . 5 ⊢ 𝒫 𝐴 ∈ V
23	22, 22	mpoex 8078	. . . 4 ⊢ (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)})) ∈ V
24	19, 20, 23	fvmpt 6986	. . 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 2782	. 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 1540   ∈ wcel 2108  ∃wrex 3060  {crab 3415  Vcvv 3459   ∪ cun 3924  𝒫 cpw 4575   class class class wbr 5119  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407  lecple 17278  joincjn 18323  Atomscatm 39281  +_𝑃cpadd 39814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-padd 39815

This theorem is referenced by:  paddval  39817