Theorem mclspps 34178

Description: The closure is closed under application of provable pre-statements. (Compare mclsax 34163.) 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 34126	. . . 4 ⊢ (𝑆 ∈ ran 𝐿 → 𝑆:𝐸⟶𝐸)
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → 𝑆:𝐸⟶𝐸)
6	5	ffnd 6669	. 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 2736	. . . . . . . . 9 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
11		mclspps.j	. . . . . . . . 9 ⊢ 𝐽 = (mPPSt‘𝑇)
12	10, 11	mppspst 34168	. . . . . . . 8 ⊢ 𝐽 ⊆ (mPreSt‘𝑇)
13		mclspps.4	. . . . . . . 8 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)
14	12, 13	sselid 3942	. . . . . . 7 ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇))
15	7, 3, 10	elmpst 34130	. . . . . . 7 ⊢ (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑀 ⊆ 𝐷 ∧ ◡𝑀 = 𝑀) ∧ (𝑂 ⊆ 𝐸 ∧ 𝑂 ∈ Fin) ∧ 𝑃 ∈ 𝐸))
16	14, 15	sylib 217	. . . . . 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 495	. . . 4 ⊢ (𝜑 → 𝑀 ⊆ 𝐷)
19	16	simp2d 1143	. . . . 5 ⊢ (𝜑 → (𝑂 ⊆ 𝐸 ∧ 𝑂 ∈ Fin))
20	19	simpld 495	. . . 4 ⊢ (𝜑 → 𝑂 ⊆ 𝐸)
21		eqid 2736	. . . 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 3143	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑂 (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))
27	5	ffund 6672	. . . . . 6 ⊢ (𝜑 → Fun 𝑆)
28	5	fdmd 6679	. . . . . . 7 ⊢ (𝜑 → dom 𝑆 = 𝐸)
29	20, 28	sseqtrrd 3985	. . . . . 6 ⊢ (𝜑 → 𝑂 ⊆ dom 𝑆)
30		funimass5 7005	. . . . . 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 256	. . . 4 ⊢ (𝜑 → 𝑂 ⊆ (◡𝑆 “ (𝐾𝐶𝐵)))
33	22, 3, 23	mvhf 34152	. . . . . . 7 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸)
34	9, 33	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻:𝑉⟶𝐸)
35	34	ffvelcdmda 7035	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ 𝐸)
36		mclspps.7	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))
37		elpreima 7008	. . . . . . 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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ((𝐻‘𝑣) ∈ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ ((𝐻‘𝑣) ∈ 𝐸 ∧ (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))))
40	35, 36, 39	mpbir2and 711	. . . 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 34177	. . . 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 34164	. . 3 ⊢ (𝜑 → (𝑀𝐶𝑂) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)))
58	10, 11, 8	elmpps 34167	. . . . 5 ⊢ (⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽 ↔ (⟨𝑀, 𝑂, 𝑃⟩ ∈ (mPreSt‘𝑇) ∧ 𝑃 ∈ (𝑀𝐶𝑂)))
59	58	simprbi 497	. . . 4 ⊢ (⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽 → 𝑃 ∈ (𝑀𝐶𝑂))
60	13, 59	syl 17	. . 3 ⊢ (𝜑 → 𝑃 ∈ (𝑀𝐶𝑂))
61	57, 60	sseldd 3945	. 2 ⊢ (𝜑 → 𝑃 ∈ (◡𝑆 “ (𝐾𝐶𝐵)))
62		elpreima 7008	. . 3 ⊢ (𝑆 Fn 𝐸 → (𝑃 ∈ (◡𝑆 “ (𝐾𝐶𝐵)) ↔ (𝑃 ∈ 𝐸 ∧ (𝑆‘𝑃) ∈ (𝐾𝐶𝐵))))
63	62	simplbda 500	. 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 205   ∧ wa 396   ∧ w3a 1087  ∀wal 1539   = wceq 1541   ∈ wcel 2106  ∀wral 3064   ∪ cun 3908   ⊆ wss 3910  ⟨cotp 4594   class class class wbr 5105   × cxp 5631  ^◡ccnv 5632  dom cdm 5633  ran crn 5634   “ cima 5636  Fun wfun 6490   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  Fincfn 8883  mVRcmvar 34055  mAxcmax 34059  mExcmex 34061  mDVcmdv 34062  mVarscmvrs 34063  mSubstcmsub 34065  mVHcmvh 34066  mPreStcmpst 34067  mFScmfs 34070  mClscmcls 34071  mPPStcmpps 34072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-word 14403  df-lsw 14451  df-concat 14459  df-s1 14484  df-substr 14529  df-pfx 14559  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-frmd 18659  df-vrmd 18660  df-mrex 34080  df-mex 34081  df-mdv 34082  df-mvrs 34083  df-mrsub 34084  df-msub 34085  df-mvh 34086  df-mpst 34087  df-msr 34088  df-msta 34089  df-mfs 34090  df-mcls 34091  df-mpps 34092

This theorem is referenced by: (None)