Theorem dfpet2parts2 39708

Description: Grade stability applied to the decomposed PetParts modules.

Pet2Parts is obtained by applying the grade-stability operator SucMap ShiftStable (see df-shiftstable 39217) to the modular intersection from dfpetparts2 39707. This makes the two orthogonal stability axes explicit:

(E) semantic stability / equilibrium: BlockLiftFix,

(G) grade stability: SucMap ShiftStable,

assembled on top of typedness and disjoint-span base modules.

This is the principled "extra level" that does not arise for Disjs: disjoint relations already bundle their internal map/carrier consistency via QMap and ElDisjs (see dfdisjs6 39677 / dfdisjs7 39678), while the present construction has an additional external grading axis imposed by the canonical successor map SucMap. (Contributed by Peter Mazsa, 20-Feb-2026.) (Revised by Peter Mazsa, 25-Feb-2026.)

Assertion

Ref	Expression
dfpet2parts2	⊢ Pet2Parts = ( SucMap ShiftStable ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix ))

Distinct variable group:   𝑛,𝑟

Proof of Theorem dfpet2parts2

Step	Hyp	Ref	Expression
1		df-pet2parts 39705	. 2 ⊢ Pet2Parts = ( SucMap ShiftStable PetParts )
2		dfpetparts2 39707	. . 3 ⊢ PetParts = ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix )
3		shiftstableeq2 39218	. . 3 ⊢ ( PetParts = ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix ) → ( SucMap ShiftStable PetParts ) = ( SucMap ShiftStable ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix )))
4	2, 3	ax-mp 5	. 2 ⊢ ( SucMap ShiftStable PetParts ) = ( SucMap ShiftStable ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix ))
5	1, 4	eqtri 2785	1 ⊢ Pet2Parts = ( SucMap ShiftStable ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix ))

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145   ∩ cin 3901  {copab 5171   E cep 5558   × cxp 5657  ^◡ccnv 5658   ↾ cres 5661   ⋉ cxrn 38909   SucMap csucmap 38913   BlockLiftFix cblockliftfix 38916   ShiftStable cshiftstable 38917   Rels crels 38920   Disjs cdisjs 38953   MembParts cmembparts 38960   PetParts cpetparts 38962   Pet2Parts cpet2parts 38963

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-eprel 5559  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-1st 7989  df-2nd 7990  df-ec 8701  df-qs 8705  df-xrn 39115  df-blockliftfix 39216  df-shiftstable 39217  df-dmqss 39457  df-parts 39603  df-petparts 39703  df-pet2parts 39705

This theorem is used by: (None)