Theorem mclspps 35649

Description: The closure is closed under application of provable pre-statements. (Compare mclsax 35634.) 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 35597	. . . 4 ⊢ (𝑆 ∈ ran 𝐿 → 𝑆:𝐸⟶𝐸)
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → 𝑆:𝐸⟶𝐸)
6	5	ffnd 6657	. 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 2733	. . . . . . . . 9 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
11		mclspps.j	. . . . . . . . 9 ⊢ 𝐽 = (mPPSt‘𝑇)
12	10, 11	mppspst 35639	. . . . . . . 8 ⊢ 𝐽 ⊆ (mPreSt‘𝑇)
13		mclspps.4	. . . . . . . 8 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)
14	12, 13	sselid 3928	. . . . . . 7 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇))
15	7, 3, 10	elmpst 35601	. . . . . . 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 2733	. . . 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 3125	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑂 (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
27	5	ffund 6660	. . . . . 6 ⊢ (𝜑 → Fun 𝑆)
28	5	fdmd 6666	. . . . . . 7 ⊢ (𝜑 → dom 𝑆 = 𝐸)
29	20, 28	sseqtrrd 3968	. . . . . 6 ⊢ (𝜑 → 𝑂 ⊆ dom 𝑆)
30		funimass5 6994	. . . . . 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 35623	. . . . . . 7 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸)
34	9, 33	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻:𝑉⟶𝐸)
35	34	ffvelcdmda 7023	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ 𝐸)
36		mclspps.7	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))
37		elpreima 6997	. . . . . . 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 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 1186	. . . . 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 1186	. . . . 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 1186	. . . . 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 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)) → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇))
53		simp22 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 ∈ ran 𝐿)
54		simp23 1209	. . . . 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 35648	. . . 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 35635	. . 3 ⊢ (𝜑 → (𝑀𝐶𝑂) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)))
58	10, 11, 8	elmpps 35638	. . . . 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 3931	. 2 ⊢ (𝜑 → 𝑃 ∈ (◡𝑆 “ (𝐾𝐶𝐵)))
62		elpreima 6997	. . 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 1539   = wceq 1541   ∈ wcel 2113  ∀wral 3048   ∪ cun 3896   ⊆ wss 3898  ⟨cotp 4583   class class class wbr 5093   × cxp 5617  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   “ cima 5622  Fun wfun 6480   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486  (class class class)co 7352  Fincfn 8875  mVRcmvar 35526  mAxcmax 35530  mExcmex 35532  mDVcmdv 35533  mVarscmvrs 35534  mSubstcmsub 35536  mVHcmvh 35537  mPreStcmpst 35538  mFScmfs 35541  mClscmcls 35542  mPPStcmpps 35543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-ot 4584  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-er 8628  df-map 8758  df-pm 8759  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-2 12195  df-n0 12389  df-xnn0 12462  df-z 12476  df-uz 12739  df-fz 13410  df-fzo 13557  df-seq 13911  df-hash 14240  df-word 14423  df-lsw 14472  df-concat 14480  df-s1 14506  df-substr 14551  df-pfx 14581  df-struct 17060  df-sets 17077  df-slot 17095  df-ndx 17107  df-base 17123  df-ress 17144  df-plusg 17176  df-0g 17347  df-gsum 17348  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-mhm 18693  df-submnd 18694  df-frmd 18759  df-vrmd 18760  df-mrex 35551  df-mex 35552  df-mdv 35553  df-mvrs 35554  df-mrsub 35555  df-msub 35556  df-mvh 35557  df-mpst 35558  df-msr 35559  df-msta 35560  df-mfs 35561  df-mcls 35562  df-mpps 35563

This theorem is referenced by: (None)