Theorem mclspps 35569

Description: The closure is closed under application of provable pre-statements. (Compare mclsax 35554.) 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 35517	. . . 4 ⊢ (𝑆 ∈ ran 𝐿 → 𝑆:𝐸⟶𝐸)
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → 𝑆:𝐸⟶𝐸)
6	5	ffnd 6738	. 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 2735	. . . . . . . . 9 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
11		mclspps.j	. . . . . . . . 9 ⊢ 𝐽 = (mPPSt‘𝑇)
12	10, 11	mppspst 35559	. . . . . . . 8 ⊢ 𝐽 ⊆ (mPreSt‘𝑇)
13		mclspps.4	. . . . . . . 8 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)
14	12, 13	sselid 3993	. . . . . . 7 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇))
15	7, 3, 10	elmpst 35521	. . . . . . 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 1141	. . . . 5 ⊢ (𝜑 → (𝑀 ⊆ 𝐷 ∧ ◡𝑀 = 𝑀))
18	17	simpld 494	. . . 4 ⊢ (𝜑 → 𝑀 ⊆ 𝐷)
19	16	simp2d 1142	. . . . 5 ⊢ (𝜑 → (𝑂 ⊆ 𝐸 ∧ 𝑂 ∈ Fin))
20	19	simpld 494	. . . 4 ⊢ (𝜑 → 𝑂 ⊆ 𝐸)
21		eqid 2735	. . . 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 3144	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑂 (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
27	5	ffund 6741	. . . . . 6 ⊢ (𝜑 → Fun 𝑆)
28	5	fdmd 6747	. . . . . . 7 ⊢ (𝜑 → dom 𝑆 = 𝐸)
29	20, 28	sseqtrrd 4037	. . . . . 6 ⊢ (𝜑 → 𝑂 ⊆ dom 𝑆)
30		funimass5 7075	. . . . . 6 ⊢ ((Fun 𝑆 ∧ 𝑂 ⊆ dom 𝑆) → (𝑂 ⊆ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ ∀𝑥 ∈ 𝑂 (𝑆‘𝑥) ∈ (𝐾𝐶𝐵)))
31	27, 29, 30	syl2anc 584	. . . . 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 35543	. . . . . . 7 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸)
34	9, 33	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻:𝑉⟶𝐸)
35	34	ffvelcdmda 7104	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ 𝐸)
36		mclspps.7	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))
37		elpreima 7078	. . . . . . 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 713	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ (◡𝑆 “ (𝐾𝐶𝐵)))
41	9	3ad2ant1 1132	. . . . 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 1132	. . . . 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 1132	. . . . 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 1132	. . . . 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 1132	. . . . 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 1184	. . . . 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 1184	. . . . 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 1184	. . . . 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 1205	. . . . 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 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)) → 𝑠 ∈ ran 𝐿)
54		simp23 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 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)))
55		simp3 1137	. . . . 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 35568	. . . 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 35555	. . 3 ⊢ (𝜑 → (𝑀𝐶𝑂) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)))
58	10, 11, 8	elmpps 35558	. . . . 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 3996	. 2 ⊢ (𝜑 → 𝑃 ∈ (◡𝑆 “ (𝐾𝐶𝐵)))
62		elpreima 7078	. . 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 584	1 ⊢ (𝜑 → (𝑆‘𝑃) ∈ (𝐾𝐶𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1535   = wceq 1537   ∈ wcel 2106  ∀wral 3059   ∪ cun 3961   ⊆ wss 3963  ⟨cotp 4639   class class class wbr 5148   × cxp 5687  ^◡ccnv 5688  dom cdm 5689  ran crn 5690   “ cima 5692  Fun wfun 6557   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  Fincfn 8984  mVRcmvar 35446  mAxcmax 35450  mExcmex 35452  mDVcmdv 35453  mVarscmvrs 35454  mSubstcmsub 35456  mVHcmvh 35457  mPreStcmpst 35458  mFScmfs 35461  mClscmcls 35462  mPPStcmpps 35463

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-word 14550  df-lsw 14598  df-concat 14606  df-s1 14631  df-substr 14676  df-pfx 14706  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-0g 17488  df-gsum 17489  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-frmd 18875  df-vrmd 18876  df-mrex 35471  df-mex 35472  df-mdv 35473  df-mvrs 35474  df-mrsub 35475  df-msub 35476  df-mvh 35477  df-mpst 35478  df-msr 35479  df-msta 35480  df-mfs 35481  df-mcls 35482  df-mpps 35483

This theorem is referenced by: (None)