Theorem dfpet2parts2 39650

Description: Grade stability applied to the decomposed PetParts modules.

Pet2Parts is obtained by applying the grade-stability operator SucMap ShiftStable (see df-shiftstable 39159) to the modular intersection from dfpetparts2 39649. 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 39619 / dfdisjs7 39620), 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 39647	. 2 ⊢ Pet2Parts = ( SucMap ShiftStable PetParts )
2		dfpetparts2 39649	. . 3 ⊢ PetParts = ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix )
3		shiftstableeq2 39160	. . 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 1569   ∈ wcel 2142   ∩ cin 3903  {copab 5172   E cep 5559   × cxp 5658  ^◡ccnv 5659   ↾ cres 5662   ⋉ cxrn 38851   SucMap csucmap 38855   BlockLiftFix cblockliftfix 38858   ShiftStable cshiftstable 38859   Rels crels 38862   Disjs cdisjs 38895   MembParts cmembparts 38902   PetParts cpetparts 38904   Pet2Parts cpet2parts 38905

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  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 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-eprel 5560  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-1st 7984  df-2nd 7985  df-ec 8694  df-qs 8698  df-xrn 39057  df-blockliftfix 39158  df-shiftstable 39159  df-dmqss 39399  df-parts 39545  df-petparts 39645  df-pet2parts 39647

This theorem is used by: (None)