Theorem mclspps 35552

Description: The closure is closed under application of provable pre-statements. (Compare mclsax 35537.) 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	⊢ (𝜑 → 𝑇 ∈ mFS)
mclspps.2	⊢ (𝜑 → 𝐾 ⊆ 𝐷)
mclspps.3	⊢ (𝜑 → 𝐵 ⊆ 𝐸)
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	⊢ (𝜑 → 𝑆 ∈ ran 𝐿)
mclspps.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
mclspps.7	⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))
mclspps.8	⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏)

Assertion

Ref	Expression
mclspps	⊢ (𝜑 → (𝑆‘𝑃) ∈ (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.

Step	Hyp	Ref	Expression
1		mclspps.5	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ran 𝐿)
2		mclspps.l	. . . . 5 ⊢ 𝐿 = (mSubst‘𝑇)
3		mclspps.e	. . . . 5 ⊢ 𝐸 = (mEx‘𝑇)
4	2, 3	msubf 35500	. . . 4 ⊢ (𝑆 ∈ ran 𝐿 → 𝑆:𝐸⟶𝐸)
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → 𝑆:𝐸⟶𝐸)
6	5	ffnd 6748	. 2 ⊢ (𝜑 → 𝑆 Fn 𝐸)
7		mclspps.d	. . . 4 ⊢ 𝐷 = (mDV‘𝑇)
8		mclspps.c	. . . 4 ⊢ 𝐶 = (mCls‘𝑇)
9		mclspps.1	. . . 4 ⊢ (𝜑 → 𝑇 ∈ mFS)
10		eqid 2740	. . . . . . . . 9 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
11		mclspps.j	. . . . . . . . 9 ⊢ 𝐽 = (mPPSt‘𝑇)
12	10, 11	mppspst 35542	. . . . . . . 8 ⊢ 𝐽 ⊆ (mPreSt‘𝑇)
13		mclspps.4	. . . . . . . 8 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)
14	12, 13	sselid 4006	. . . . . . 7 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇))
15	7, 3, 10	elmpst 35504	. . . . . . 7 ⊢ (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑀 ⊆ 𝐷 ∧ ◡𝑀 = 𝑀) ∧ (𝑂 ⊆ 𝐸 ∧ 𝑂 ∈ Fin) ∧ 𝑃 ∈ 𝐸))
16	14, 15	sylib 218	. . . . . 6 ⊢ (𝜑 → ((𝑀 ⊆ 𝐷 ∧ ◡𝑀 = 𝑀) ∧ (𝑂 ⊆ 𝐸 ∧ 𝑂 ∈ Fin) ∧ 𝑃 ∈ 𝐸))
17	16	simp1d 1142	. . . . 5 ⊢ (𝜑 → (𝑀 ⊆ 𝐷 ∧ ◡𝑀 = 𝑀))
18	17	simpld 494	. . . 4 ⊢ (𝜑 → 𝑀 ⊆ 𝐷)
19	16	simp2d 1143	. . . . 5 ⊢ (𝜑 → (𝑂 ⊆ 𝐸 ∧ 𝑂 ∈ Fin))
20	19	simpld 494	. . . 4 ⊢ (𝜑 → 𝑂 ⊆ 𝐸)
21		eqid 2740	. . . 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 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
26	25	ralrimiva 3152	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑂 (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
27	5	ffund 6751	. . . . . 6 ⊢ (𝜑 → Fun 𝑆)
28	5	fdmd 6757	. . . . . . 7 ⊢ (𝜑 → dom 𝑆 = 𝐸)
29	20, 28	sseqtrrd 4050	. . . . . 6 ⊢ (𝜑 → 𝑂 ⊆ dom 𝑆)
30		funimass5 7088	. . . . . 6 ⊢ ((Fun 𝑆 ∧ 𝑂 ⊆ dom 𝑆) → (𝑂 ⊆ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ ∀𝑥 ∈ 𝑂 (𝑆‘𝑥) ∈ (𝐾𝐶𝐵)))
31	27, 29, 30	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝑂 ⊆ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ ∀𝑥 ∈ 𝑂 (𝑆‘𝑥) ∈ (𝐾𝐶𝐵)))
32	26, 31	mpbird 257	. . . 4 ⊢ (𝜑 → 𝑂 ⊆ (◡𝑆 “ (𝐾𝐶𝐵)))
33	22, 3, 23	mvhf 35526	. . . . . . 7 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸)
34	9, 33	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻:𝑉⟶𝐸)
35	34	ffvelcdmda 7118	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ 𝐸)
36		mclspps.7	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))
37		elpreima 7091	. . . . . . 7 ⊢ (𝑆 Fn 𝐸 → ((𝐻‘𝑣) ∈ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝐻‘𝑣) ∈ 𝐸 ∧ (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))))
38	6, 37	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝐻‘𝑣) ∈ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝐻‘𝑣) ∈ 𝐸 ∧ (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))))
39	38	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ((𝐻‘𝑣) ∈ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝐻‘𝑣) ∈ 𝐸 ∧ (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))))
40	35, 36, 39	mpbir2and 712	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ (◡𝑆 “ (𝐾𝐶𝐵)))
41	9	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) → 𝑇 ∈ mFS)
42		mclspps.2	. . . . . 6 ⊢ (𝜑 → 𝐾 ⊆ 𝐷)
43	42	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) → 𝐾 ⊆ 𝐷)
44		mclspps.3	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐸)
45	44	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) → 𝐵 ⊆ 𝐸)
46	13	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)
47	1	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) → 𝑆 ∈ ran 𝐿)
48	25	3ad2antl1 1185	. . . . 5 ⊢ (((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) ∧ 𝑥 ∈ 𝑂) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
49	36	3ad2antl1 1185	. . . . 5 ⊢ (((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))
50		mclspps.8	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏)
51	50	3ad2antl1 1185	. . . . 5 ⊢ (((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏)
52		simp21 1206	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇))
53		simp22 1207	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) → 𝑠 ∈ ran 𝐿)
54		simp23 1208	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)))
55		simp3 1138	. . . . 5 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) → ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀))
56	7, 3, 8, 41, 43, 45, 11, 2, 22, 23, 24, 46, 47, 48, 49, 51, 52, 53, 54, 55	mclsppslem 35551	. . . 4 ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) ∧ ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) → (𝑠‘𝑝) ∈ (◡𝑆 “ (𝐾𝐶𝐵)))
57	7, 3, 8, 9, 18, 20, 21, 2, 22, 23, 24, 32, 40, 56	mclsind 35538	. . 3 ⊢ (𝜑 → (𝑀𝐶𝑂) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)))
58	10, 11, 8	elmpps 35541	. . . . 5 ⊢ (⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽 ↔ (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) ∧ 𝑃 ∈ (𝑀𝐶𝑂)))
59	58	simprbi 496	. . . 4 ⊢ (⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽 → 𝑃 ∈ (𝑀𝐶𝑂))
60	13, 59	syl 17	. . 3 ⊢ (𝜑 → 𝑃 ∈ (𝑀𝐶𝑂))
61	57, 60	sseldd 4009	. 2 ⊢ (𝜑 → 𝑃 ∈ (◡𝑆 “ (𝐾𝐶𝐵)))
62		elpreima 7091	. . 3 ⊢ (𝑆 Fn 𝐸 → (𝑃 ∈ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ (𝑃 ∈ 𝐸 ∧ (𝑆‘𝑃) ∈ (𝐾𝐶𝐵))))
63	62	simplbda 499	. 2 ⊢ ((𝑆 Fn 𝐸 ∧ 𝑃 ∈ (◡𝑆 “ (𝐾𝐶𝐵))) → (𝑆‘𝑃) ∈ (𝐾𝐶𝐵))
64	6, 61, 63	syl2anc 583	1 ⊢ (𝜑 → (𝑆‘𝑃) ∈ (𝐾𝐶𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1535   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ∪ cun 3974   ⊆ wss 3976  ⟨cotp 4656   class class class wbr 5166   × cxp 5698  ^◡ccnv 5699  dom cdm 5700  ran crn 5701   “ cima 5703  Fun wfun 6567   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  Fincfn 9003  mVRcmvar 35429  mAxcmax 35433  mExcmex 35435  mDVcmdv 35436  mVarscmvrs 35437  mSubstcmsub 35439  mVHcmvh 35440  mPreStcmpst 35441  mFScmfs 35444  mClscmcls 35445  mPPStcmpps 35446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-seq 14053  df-hash 14380  df-word 14563  df-lsw 14611  df-concat 14619  df-s1 14644  df-substr 14689  df-pfx 14719  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-0g 17501  df-gsum 17502  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-mhm 18818  df-submnd 18819  df-frmd 18884  df-vrmd 18885  df-mrex 35454  df-mex 35455  df-mdv 35456  df-mvrs 35457  df-mrsub 35458  df-msub 35459  df-mvh 35460  df-mpst 35461  df-msr 35462  df-msta 35463  df-mfs 35464  df-mcls 35465  df-mpps 35466

This theorem is referenced by: (None)