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Theorem mclspps 31709
 Description: The closure is closed under application of provable pre-statements. (Compare mclsax 31694.) 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 31657 . . . 4 (𝑆 ∈ ran 𝐿𝑆:𝐸𝐸)
51, 4syl 17 . . 3 (𝜑𝑆:𝐸𝐸)
6 ffn 6158 . . 3 (𝑆:𝐸𝐸𝑆 Fn 𝐸)
75, 6syl 17 . 2 (𝜑𝑆 Fn 𝐸)
8 mclspps.d . . . 4 𝐷 = (mDV‘𝑇)
9 mclspps.c . . . 4 𝐶 = (mCls‘𝑇)
10 mclspps.1 . . . 4 (𝜑𝑇 ∈ mFS)
11 eqid 2724 . . . . . . . . 9 (mPreSt‘𝑇) = (mPreSt‘𝑇)
12 mclspps.j . . . . . . . . 9 𝐽 = (mPPSt‘𝑇)
1311, 12mppspst 31699 . . . . . . . 8 𝐽 ⊆ (mPreSt‘𝑇)
14 mclspps.4 . . . . . . . 8 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)
1513, 14sseldi 3707 . . . . . . 7 (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇))
168, 3, 11elmpst 31661 . . . . . . 7 (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑀𝐷𝑀 = 𝑀) ∧ (𝑂𝐸𝑂 ∈ Fin) ∧ 𝑃𝐸))
1715, 16sylib 208 . . . . . 6 (𝜑 → ((𝑀𝐷𝑀 = 𝑀) ∧ (𝑂𝐸𝑂 ∈ Fin) ∧ 𝑃𝐸))
1817simp1d 1134 . . . . 5 (𝜑 → (𝑀𝐷𝑀 = 𝑀))
1918simpld 477 . . . 4 (𝜑𝑀𝐷)
2017simp2d 1135 . . . . 5 (𝜑 → (𝑂𝐸𝑂 ∈ Fin))
2120simpld 477 . . . 4 (𝜑𝑂𝐸)
22 eqid 2724 . . . 4 (mAx‘𝑇) = (mAx‘𝑇)
23 mclspps.v . . . 4 𝑉 = (mVR‘𝑇)
24 mclspps.h . . . 4 𝐻 = (mVH‘𝑇)
25 mclspps.w . . . 4 𝑊 = (mVars‘𝑇)
26 mclspps.6 . . . . . 6 ((𝜑𝑥𝑂) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))
2726ralrimiva 3068 . . . . 5 (𝜑 → ∀𝑥𝑂 (𝑆𝑥) ∈ (𝐾𝐶𝐵))
28 ffun 6161 . . . . . . 7 (𝑆:𝐸𝐸 → Fun 𝑆)
295, 28syl 17 . . . . . 6 (𝜑 → Fun 𝑆)
30 fdm 6164 . . . . . . . 8 (𝑆:𝐸𝐸 → dom 𝑆 = 𝐸)
315, 30syl 17 . . . . . . 7 (𝜑 → dom 𝑆 = 𝐸)
3221, 31sseqtr4d 3748 . . . . . 6 (𝜑𝑂 ⊆ dom 𝑆)
33 funimass5 6449 . . . . . 6 ((Fun 𝑆𝑂 ⊆ dom 𝑆) → (𝑂 ⊆ (𝑆 “ (𝐾𝐶𝐵)) ↔ ∀𝑥𝑂 (𝑆𝑥) ∈ (𝐾𝐶𝐵)))
3429, 32, 33syl2anc 696 . . . . 5 (𝜑 → (𝑂 ⊆ (𝑆 “ (𝐾𝐶𝐵)) ↔ ∀𝑥𝑂 (𝑆𝑥) ∈ (𝐾𝐶𝐵)))
3527, 34mpbird 247 . . . 4 (𝜑𝑂 ⊆ (𝑆 “ (𝐾𝐶𝐵)))
3623, 3, 24mvhf 31683 . . . . . . 7 (𝑇 ∈ mFS → 𝐻:𝑉𝐸)
3710, 36syl 17 . . . . . 6 (𝜑𝐻:𝑉𝐸)
3837ffvelrnda 6474 . . . . 5 ((𝜑𝑣𝑉) → (𝐻𝑣) ∈ 𝐸)
39 mclspps.7 . . . . 5 ((𝜑𝑣𝑉) → (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))
40 elpreima 6452 . . . . . . 7 (𝑆 Fn 𝐸 → ((𝐻𝑣) ∈ (𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝐻𝑣) ∈ 𝐸 ∧ (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))))
417, 40syl 17 . . . . . 6 (𝜑 → ((𝐻𝑣) ∈ (𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝐻𝑣) ∈ 𝐸 ∧ (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))))
4241adantr 472 . . . . 5 ((𝜑𝑣𝑉) → ((𝐻𝑣) ∈ (𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝐻𝑣) ∈ 𝐸 ∧ (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))))
4338, 39, 42mpbir2and 995 . . . 4 ((𝜑𝑣𝑉) → (𝐻𝑣) ∈ (𝑆 “ (𝐾𝐶𝐵)))
44103ad2ant1 1125 . . . . 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)
45 mclspps.2 . . . . . 6 (𝜑𝐾𝐷)
46453ad2ant1 1125 . . . . 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)
47 mclspps.3 . . . . . 6 (𝜑𝐵𝐸)
48473ad2ant1 1125 . . . . 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)
49143ad2ant1 1125 . . . . 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)
5013ad2ant1 1125 . . . . 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)
51263ad2antl1 1177 . . . . 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))
52393ad2antl1 1177 . . . . 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))
53 mclspps.8 . . . . . 6 ((𝜑 ∧ (𝑥𝑀𝑦𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))))) → 𝑎𝐾𝑏)
54533ad2antl1 1177 . . . . 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)
55 simp21 1225 . . . . 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))
56 simp22 1226 . . . . 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)
57 simp23 1227 . . . . 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)))
58 simp3 1130 . . . . 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))
598, 3, 9, 44, 46, 48, 12, 2, 23, 24, 25, 49, 50, 51, 52, 54, 55, 56, 57, 58mclsppslem 31708 . . . 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)))
608, 3, 9, 10, 19, 21, 22, 2, 23, 24, 25, 35, 43, 59mclsind 31695 . . 3 (𝜑 → (𝑀𝐶𝑂) ⊆ (𝑆 “ (𝐾𝐶𝐵)))
6111, 12, 9elmpps 31698 . . . . 5 (⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽 ↔ (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) ∧ 𝑃 ∈ (𝑀𝐶𝑂)))
6261simprbi 483 . . . 4 (⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽𝑃 ∈ (𝑀𝐶𝑂))
6314, 62syl 17 . . 3 (𝜑𝑃 ∈ (𝑀𝐶𝑂))
6460, 63sseldd 3710 . 2 (𝜑𝑃 ∈ (𝑆 “ (𝐾𝐶𝐵)))
65 elpreima 6452 . . 3 (𝑆 Fn 𝐸 → (𝑃 ∈ (𝑆 “ (𝐾𝐶𝐵)) ↔ (𝑃𝐸 ∧ (𝑆𝑃) ∈ (𝐾𝐶𝐵))))
6665simplbda 655 . 2 ((𝑆 Fn 𝐸𝑃 ∈ (𝑆 “ (𝐾𝐶𝐵))) → (𝑆𝑃) ∈ (𝐾𝐶𝐵))
677, 64, 66syl2anc 696 1 (𝜑 → (𝑆𝑃) ∈ (𝐾𝐶𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072  ∀wal 1594   = wceq 1596   ∈ wcel 2103  ∀wral 3014   ∪ cun 3678   ⊆ wss 3680  ⟨cotp 4293   class class class wbr 4760   × cxp 5216  ◡ccnv 5217  dom cdm 5218  ran crn 5219   “ cima 5221  Fun wfun 5995   Fn wfn 5996  ⟶wf 5997  ‘cfv 6001  (class class class)co 6765  Fincfn 8072  mVRcmvar 31586  mAxcmax 31590  mExcmex 31592  mDVcmdv 31593  mVarscmvrs 31594  mSubstcmsub 31596  mVHcmvh 31597  mPreStcmpst 31598  mFScmfs 31601  mClscmcls 31602  mPPStcmpps 31603 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-ot 4294  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-er 7862  df-map 7976  df-pm 7977  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-card 8878  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-2 11192  df-n0 11406  df-xnn0 11477  df-z 11491  df-uz 11801  df-fz 12441  df-fzo 12581  df-seq 12917  df-hash 13233  df-word 13406  df-lsw 13407  df-concat 13408  df-s1 13409  df-substr 13410  df-struct 15982  df-ndx 15983  df-slot 15984  df-base 15986  df-sets 15987  df-ress 15988  df-plusg 16077  df-0g 16225  df-gsum 16226  df-mgm 17364  df-sgrp 17406  df-mnd 17417  df-mhm 17457  df-submnd 17458  df-frmd 17508  df-vrmd 17509  df-mrex 31611  df-mex 31612  df-mdv 31613  df-mvrs 31614  df-mrsub 31615  df-msub 31616  df-mvh 31617  df-mpst 31618  df-msr 31619  df-msta 31620  df-mfs 31621  df-mcls 31622  df-mpps 31623 This theorem is referenced by: (None)
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