Theorem dfpet2parts2 39176

Description: Grade stability applied to the decomposed PetParts modules.

Pet2Parts is obtained by applying the grade-stability operator SucMap ShiftStable (see df-shiftstable 38685) to the modular intersection from dfpetparts2 39175. 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 39145 / dfdisjs7 39146), 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 39173	. 2 ⊢ Pet2Parts = ( SucMap ShiftStable PetParts )
2		dfpetparts2 39175	. . 3 ⊢ PetParts = ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix )
3		shiftstableeq2 38686	. . 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 2760	1 ⊢ Pet2Parts = ( SucMap ShiftStable ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix ))

Syntax hints:   = wceq 1542   ∈ wcel 2114   ∩ cin 3901  {copab 5161   E cep 5524   × cxp 5623  ^◡ccnv 5624   ↾ cres 5627   ⋉ cxrn 38377   SucMap csucmap 38381   BlockLiftFix cblockliftfix 38384   ShiftStable cshiftstable 38385   Rels crels 38388   Disjs cdisjs 38421   MembParts cmembparts 38428   PetParts cpetparts 38430   Pet2Parts cpet2parts 38431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-eprel 5525  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499  df-fv 6501  df-1st 7935  df-2nd 7936  df-ec 8639  df-qs 8643  df-xrn 38583  df-blockliftfix 38684  df-shiftstable 38685  df-dmqss 38925  df-parts 39071  df-petparts 39171  df-pet2parts 39173

This theorem is referenced by: (None)