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Theorem mclspps 32826
Description: The closure is closed under application of provable pre-statements. (Compare mclsax 32811.) This theorem is what justifies the treatment of theorems as "equivalent" to axioms once they have been proven: the composition of one theorem in the proof of another yields a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mclspps.d 𝐷 = (mDV‘𝑇)
mclspps.e 𝐸 = (mEx‘𝑇)
mclspps.c 𝐶 = (mCls‘𝑇)
mclspps.1 (𝜑𝑇 ∈ mFS)
mclspps.2 (𝜑𝐾𝐷)
mclspps.3 (𝜑𝐵𝐸)
mclspps.j 𝐽 = (mPPSt‘𝑇)
mclspps.l 𝐿 = (mSubst‘𝑇)
mclspps.v 𝑉 = (mVR‘𝑇)
mclspps.h 𝐻 = (mVH‘𝑇)
mclspps.w 𝑊 = (mVars‘𝑇)
mclspps.4 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)
mclspps.5 (𝜑𝑆 ∈ ran 𝐿)
mclspps.6 ((𝜑𝑥𝑂) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
mclspps.7 ((𝜑𝑣𝑉) → (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))
mclspps.8 ((𝜑 ∧ (𝑥𝑀𝑦𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))))) → 𝑎𝐾𝑏)
Assertion
Ref Expression
mclspps (𝜑 → (𝑆𝑃) ∈ (𝐾𝐶𝐵))
Distinct variable groups:   𝑣,𝐸   𝑎,𝑏,𝑣,𝑥,𝑦,𝐻   𝑣,𝑉   𝐾,𝑎,𝑏,𝑣,𝑥,𝑦   𝑇,𝑎,𝑏,𝑣,𝑥,𝑦   𝐿,𝑎,𝑏,𝑣,𝑥,𝑦   𝑆,𝑎,𝑏,𝑣,𝑥,𝑦   𝐵,𝑎,𝑏,𝑣,𝑥,𝑦   𝑊,𝑎,𝑏,𝑣,𝑥,𝑦   𝐶,𝑎,𝑏,𝑣,𝑥,𝑦   𝑀,𝑎,𝑏,𝑣,𝑥,𝑦   𝑣,𝑂,𝑥   𝜑,𝑎,𝑏,𝑣,𝑥,𝑦
Allowed substitution hints:   𝐷(𝑥,𝑦,𝑣,𝑎,𝑏)   𝑃(𝑥,𝑦,𝑣,𝑎,𝑏)   𝐸(𝑥,𝑦,𝑎,𝑏)   𝐽(𝑥,𝑦,𝑣,𝑎,𝑏)   𝑂(𝑦,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem mclspps
Dummy variables 𝑚 𝑜 𝑝 𝑠 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mclspps.5 . . . 4 (𝜑𝑆 ∈ ran 𝐿)
2 mclspps.l . . . . 5 𝐿 = (mSubst‘𝑇)
3 mclspps.e . . . . 5 𝐸 = (mEx‘𝑇)
42, 3msubf 32774 . . . 4 (𝑆 ∈ ran 𝐿𝑆:𝐸𝐸)
51, 4syl 17 . . 3 (𝜑𝑆:𝐸𝐸)
65ffnd 6510 . 2 (𝜑𝑆 Fn 𝐸)
7 mclspps.d . . . 4 𝐷 = (mDV‘𝑇)
8 mclspps.c . . . 4 𝐶 = (mCls‘𝑇)
9 mclspps.1 . . . 4 (𝜑𝑇 ∈ mFS)
10 eqid 2821 . . . . . . . . 9 (mPreSt‘𝑇) = (mPreSt‘𝑇)
11 mclspps.j . . . . . . . . 9 𝐽 = (mPPSt‘𝑇)
1210, 11mppspst 32816 . . . . . . . 8 𝐽 ⊆ (mPreSt‘𝑇)
13 mclspps.4 . . . . . . . 8 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)
1412, 13sseldi 3965 . . . . . . 7 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇))
157, 3, 10elmpst 32778 . . . . . . 7 (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑀𝐷𝑀 = 𝑀) ∧ (𝑂𝐸𝑂 ∈ Fin) ∧ 𝑃𝐸))
1614, 15sylib 220 . . . . . 6 (𝜑 → ((𝑀𝐷𝑀 = 𝑀) ∧ (𝑂𝐸𝑂 ∈ Fin) ∧ 𝑃𝐸))
1716simp1d 1138 . . . . 5 (𝜑 → (𝑀𝐷𝑀 = 𝑀))
1817simpld 497 . . . 4 (𝜑𝑀𝐷)
1916simp2d 1139 . . . . 5 (𝜑 → (𝑂𝐸𝑂 ∈ Fin))
2019simpld 497 . . . 4 (𝜑𝑂𝐸)
21 eqid 2821 . . . 4 (mAx‘𝑇) = (mAx‘𝑇)
22 mclspps.v . . . 4 𝑉 = (mVR‘𝑇)
23 mclspps.h . . . 4 𝐻 = (mVH‘𝑇)
24 mclspps.w . . . 4 𝑊 = (mVars‘𝑇)
25 mclspps.6 . . . . . 6 ((𝜑𝑥𝑂) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
2625ralrimiva 3182 . . . . 5 (𝜑 → ∀𝑥𝑂 (𝑆𝑥) ∈ (𝐾𝐶𝐵))
275ffund 6513 . . . . . 6 (𝜑 → Fun 𝑆)
285fdmd 6518 . . . . . . 7 (𝜑 → dom 𝑆 = 𝐸)
2920, 28sseqtrrd 4008 . . . . . 6 (𝜑𝑂 ⊆ dom 𝑆)
30 funimass5 6820 . . . . . 6 ((Fun 𝑆𝑂 ⊆ dom 𝑆) → (𝑂 ⊆ (𝑆 “ (𝐾𝐶𝐵)) ↔ ∀𝑥𝑂 (𝑆𝑥) ∈ (𝐾𝐶𝐵)))
3127, 29, 30syl2anc 586 . . . . 5 (𝜑 → (𝑂 ⊆ (𝑆 “ (𝐾𝐶𝐵)) ↔ ∀𝑥𝑂 (𝑆𝑥) ∈ (𝐾𝐶𝐵)))
3226, 31mpbird 259 . . . 4 (𝜑𝑂 ⊆ (𝑆 “ (𝐾𝐶𝐵)))
3322, 3, 23mvhf 32800 . . . . . . 7 (𝑇 ∈ mFS → 𝐻:𝑉𝐸)
349, 33syl 17 . . . . . 6 (𝜑𝐻:𝑉𝐸)
3534ffvelrnda 6846 . . . . 5 ((𝜑𝑣𝑉) → (𝐻𝑣) ∈ 𝐸)
36 mclspps.7 . . . . 5 ((𝜑𝑣𝑉) → (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))
37 elpreima 6823 . . . . . . 7 (𝑆 Fn 𝐸 → ((𝐻𝑣) ∈ (𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝐻𝑣) ∈ 𝐸 ∧ (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))))
386, 37syl 17 . . . . . 6 (𝜑 → ((𝐻𝑣) ∈ (𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝐻𝑣) ∈ 𝐸 ∧ (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))))
3938adantr 483 . . . . 5 ((𝜑𝑣𝑉) → ((𝐻𝑣) ∈ (𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝐻𝑣) ∈ 𝐸 ∧ (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))))
4035, 36, 39mpbir2and 711 . . . 4 ((𝜑𝑣𝑉) → (𝐻𝑣) ∈ (𝑆 “ (𝐾𝐶𝐵)))
4193ad2ant1 1129 . . . . 5 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻𝑧))) × (𝑊‘(𝑠‘(𝐻𝑤)))) ⊆ 𝑀)) → 𝑇 ∈ mFS)
42 mclspps.2 . . . . . 6 (𝜑𝐾𝐷)
43423ad2ant1 1129 . . . . 5 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻𝑧))) × (𝑊‘(𝑠‘(𝐻𝑤)))) ⊆ 𝑀)) → 𝐾𝐷)
44 mclspps.3 . . . . . 6 (𝜑𝐵𝐸)
45443ad2ant1 1129 . . . . 5 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻𝑧))) × (𝑊‘(𝑠‘(𝐻𝑤)))) ⊆ 𝑀)) → 𝐵𝐸)
46133ad2ant1 1129 . . . . 5 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻𝑧))) × (𝑊‘(𝑠‘(𝐻𝑤)))) ⊆ 𝑀)) → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)
4713ad2ant1 1129 . . . . 5 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻𝑧))) × (𝑊‘(𝑠‘(𝐻𝑤)))) ⊆ 𝑀)) → 𝑆 ∈ ran 𝐿)
48253ad2antl1 1181 . . . . 5 (((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻𝑧))) × (𝑊‘(𝑠‘(𝐻𝑤)))) ⊆ 𝑀)) ∧ 𝑥𝑂) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
49363ad2antl1 1181 . . . . 5 (((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻𝑧))) × (𝑊‘(𝑠‘(𝐻𝑤)))) ⊆ 𝑀)) ∧ 𝑣𝑉) → (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))
50 mclspps.8 . . . . . 6 ((𝜑 ∧ (𝑥𝑀𝑦𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))))) → 𝑎𝐾𝑏)
51503ad2antl1 1181 . . . . 5 (((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻𝑧))) × (𝑊‘(𝑠‘(𝐻𝑤)))) ⊆ 𝑀)) ∧ (𝑥𝑀𝑦𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))))) → 𝑎𝐾𝑏)
52 simp21 1202 . . . . 5 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻𝑧))) × (𝑊‘(𝑠‘(𝐻𝑤)))) ⊆ 𝑀)) → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇))
53 simp22 1203 . . . . 5 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻𝑧))) × (𝑊‘(𝑠‘(𝐻𝑤)))) ⊆ 𝑀)) → 𝑠 ∈ ran 𝐿)
54 simp23 1204 . . . . 5 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻𝑧))) × (𝑊‘(𝑠‘(𝐻𝑤)))) ⊆ 𝑀)) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝑆 “ (𝐾𝐶𝐵)))
55 simp3 1134 . . . . 5 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻𝑧))) × (𝑊‘(𝑠‘(𝐻𝑤)))) ⊆ 𝑀)) → ∀𝑧𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻𝑧))) × (𝑊‘(𝑠‘(𝐻𝑤)))) ⊆ 𝑀))
567, 3, 8, 41, 43, 45, 11, 2, 22, 23, 24, 46, 47, 48, 49, 51, 52, 53, 54, 55mclsppslem 32825 . . . 4 ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻𝑧))) × (𝑊‘(𝑠‘(𝐻𝑤)))) ⊆ 𝑀)) → (𝑠𝑝) ∈ (𝑆 “ (𝐾𝐶𝐵)))
577, 3, 8, 9, 18, 20, 21, 2, 22, 23, 24, 32, 40, 56mclsind 32812 . . 3 (𝜑 → (𝑀𝐶𝑂) ⊆ (𝑆 “ (𝐾𝐶𝐵)))
5810, 11, 8elmpps 32815 . . . . 5 (⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽 ↔ (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) ∧ 𝑃 ∈ (𝑀𝐶𝑂)))
5958simprbi 499 . . . 4 (⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽𝑃 ∈ (𝑀𝐶𝑂))
6013, 59syl 17 . . 3 (𝜑𝑃 ∈ (𝑀𝐶𝑂))
6157, 60sseldd 3968 . 2 (𝜑𝑃 ∈ (𝑆 “ (𝐾𝐶𝐵)))
62 elpreima 6823 . . 3 (𝑆 Fn 𝐸 → (𝑃 ∈ (𝑆 “ (𝐾𝐶𝐵)) ↔ (𝑃𝐸 ∧ (𝑆𝑃) ∈ (𝐾𝐶𝐵))))
6362simplbda 502 . 2 ((𝑆 Fn 𝐸𝑃 ∈ (𝑆 “ (𝐾𝐶𝐵))) → (𝑆𝑃) ∈ (𝐾𝐶𝐵))
646, 61, 63syl2anc 586 1 (𝜑 → (𝑆𝑃) ∈ (𝐾𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083  wal 1531   = wceq 1533  wcel 2110  wral 3138  cun 3934  wss 3936  cotp 4569   class class class wbr 5059   × cxp 5548  ccnv 5549  dom cdm 5550  ran crn 5551  cima 5553  Fun wfun 6344   Fn wfn 6345  wf 6346  cfv 6350  (class class class)co 7150  Fincfn 8503  mVRcmvar 32703  mAxcmax 32707  mExcmex 32709  mDVcmdv 32710  mVarscmvrs 32711  mSubstcmsub 32713  mVHcmvh 32714  mPreStcmpst 32715  mFScmfs 32718  mClscmcls 32719  mPPStcmpps 32720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-ot 4570  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-map 8402  df-pm 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-n0 11892  df-xnn0 11962  df-z 11976  df-uz 12238  df-fz 12887  df-fzo 13028  df-seq 13364  df-hash 13685  df-word 13856  df-lsw 13909  df-concat 13917  df-s1 13944  df-substr 13997  df-pfx 14027  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-plusg 16572  df-0g 16709  df-gsum 16710  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-mhm 17950  df-submnd 17951  df-frmd 18008  df-vrmd 18009  df-mrex 32728  df-mex 32729  df-mdv 32730  df-mvrs 32731  df-mrsub 32732  df-msub 32733  df-mvh 32734  df-mpst 32735  df-msr 32736  df-msta 32737  df-mfs 32738  df-mcls 32739  df-mpps 32740
This theorem is referenced by: (None)
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