Theorem paddval 40461

Description: Projective subspace sum operation value. (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
paddval	⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) = ((𝑋 ∪ 𝑌) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)}))

Distinct variable groups:   𝐴,𝑝   𝑞,𝑝,𝑟,𝐾   𝑋,𝑝,𝑞   𝑌,𝑝,𝑞,𝑟

Allowed substitution hints:   𝐴(𝑟,𝑞)   𝐵(𝑟,𝑞,𝑝)   + (𝑟,𝑞,𝑝)   ∨ (𝑟,𝑞,𝑝)   ≤ (𝑟,𝑞,𝑝)   𝑋(𝑟)

Proof of Theorem paddval

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		biid 264	. 2 ⊢ (𝐾 ∈ 𝐵 ↔ 𝐾 ∈ 𝐵)
2		paddfval.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
3	2	fvexi 6896	. . 3 ⊢ 𝐴 ∈ V
4	3	elpw2 5305	. 2 ⊢ (𝑋 ∈ 𝒫 𝐴 ↔ 𝑋 ⊆ 𝐴)
5	3	elpw2 5305	. 2 ⊢ (𝑌 ∈ 𝒫 𝐴 ↔ 𝑌 ⊆ 𝐴)
6		paddfval.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
7		paddfval.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
8		paddfval.p	. . . . . 6 ⊢ + = (+𝑃‘𝐾)
9	6, 7, 2, 8	paddfval 40460	. . . . 5 ⊢ (𝐾 ∈ 𝐵 → + = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)})))
10	9	oveqd 7428	. . . 4 ⊢ (𝐾 ∈ 𝐵 → (𝑋 + 𝑌) = (𝑋(𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)}))𝑌))
11	10	3ad2ant1 1149	. . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → (𝑋 + 𝑌) = (𝑋(𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)}))𝑌))
12		simpl 487	. . . . . 6 ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → 𝑋 ∈ 𝒫 𝐴)
13		simpr 489	. . . . . 6 ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → 𝑌 ∈ 𝒫 𝐴)
14		unexg 7741	. . . . . . 7 ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → (𝑋 ∪ 𝑌) ∈ V)
15	3	rabex 5310	. . . . . . 7 ⊢ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)} ∈ V
16		unexg 7741	. . . . . . 7 ⊢ (((𝑋 ∪ 𝑌) ∈ V ∧ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)} ∈ V) → ((𝑋 ∪ 𝑌) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)}) ∈ V)
17	14, 15, 16	sylancl 597	. . . . . 6 ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → ((𝑋 ∪ 𝑌) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)}) ∈ V)
18	12, 13, 17	3jca 1144	. . . . 5 ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → (𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴 ∧ ((𝑋 ∪ 𝑌) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)}) ∈ V))
19	18	3adant1 1146	. . . 4 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → (𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴 ∧ ((𝑋 ∪ 𝑌) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)}) ∈ V))
20		uneq1 4123	. . . . . 6 ⊢ (𝑚 = 𝑋 → (𝑚 ∪ 𝑛) = (𝑋 ∪ 𝑛))
21		rexeq 3325	. . . . . . 7 ⊢ (𝑚 = 𝑋 → (∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟) ↔ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)))
22	21	rabbidv 3430	. . . . . 6 ⊢ (𝑚 = 𝑋 → {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)} = {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)})
23	20, 22	uneq12d 4131	. . . . 5 ⊢ (𝑚 = 𝑋 → ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)}) = ((𝑋 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)}))
24		uneq2 4124	. . . . . 6 ⊢ (𝑛 = 𝑌 → (𝑋 ∪ 𝑛) = (𝑋 ∪ 𝑌))
25		rexeq 3325	. . . . . . . 8 ⊢ (𝑛 = 𝑌 → (∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟) ↔ ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)))
26	25	rexbidv 3195	. . . . . . 7 ⊢ (𝑛 = 𝑌 → (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟) ↔ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)))
27	26	rabbidv 3430	. . . . . 6 ⊢ (𝑛 = 𝑌 → {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)} = {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)})
28	24, 27	uneq12d 4131	. . . . 5 ⊢ (𝑛 = 𝑌 → ((𝑋 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)}) = ((𝑋 ∪ 𝑌) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)}))
29		eqid 2769	. . . . 5 ⊢ (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)})) = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)}))
30	23, 28, 29	ovmpog 7570	. . . 4 ⊢ ((𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴 ∧ ((𝑋 ∪ 𝑌) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)}) ∈ V) → (𝑋(𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)}))𝑌) = ((𝑋 ∪ 𝑌) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)}))
31	19, 30	syl 18	. . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → (𝑋(𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)}))𝑌) = ((𝑋 ∪ 𝑌) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)}))
32	11, 31	eqtrd 2804	. 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝒫 𝐴 ∧ 𝑌 ∈ 𝒫 𝐴) → (𝑋 + 𝑌) = ((𝑋 ∪ 𝑌) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)}))
33	1, 4, 5, 32	syl3anbr 1178	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) = ((𝑋 ∪ 𝑌) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)}))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  {crab 3423  Vcvv 3463   ∪ cun 3911   ⊆ wss 3913  𝒫 cpw 4567   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  lecple 17316  joincjn 18366  Atomscatm 39926  +_𝑃cpadd 40458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7985  df-2nd 7986  df-padd 40459

This theorem is referenced by:  elpadd  40462  paddunssN  40471  paddcom  40476  paddssat  40477  sspadd1  40478  sspadd2  40479