Theorem mclsssvlem 35705

Description: Lemma for mclsssv 35707. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mclsval.d	⊢ 𝐷 = (mDV‘𝑇)
mclsval.e	⊢ 𝐸 = (mEx‘𝑇)
mclsval.c	⊢ 𝐶 = (mCls‘𝑇)
mclsval.1	⊢ (𝜑 → 𝑇 ∈ mFS)
mclsval.2	⊢ (𝜑 → 𝐾 ⊆ 𝐷)
mclsval.3	⊢ (𝜑 → 𝐵 ⊆ 𝐸)
mclsval.h	⊢ 𝐻 = (mVH‘𝑇)
mclsval.a	⊢ 𝐴 = (mAx‘𝑇)
mclsval.s	⊢ 𝑆 = (mSubst‘𝑇)
mclsval.v	⊢ 𝑉 = (mVars‘𝑇)

Assertion

Ref	Expression
mclsssvlem	⊢ (𝜑 → ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ 𝐸)

Distinct variable groups:   𝑚,𝑐,𝑜,𝑝,𝑠,𝐸   𝑥,𝑐,𝐻,𝑚,𝑜,𝑝,𝑠   𝑦,𝑐,𝐵,𝑚,𝑜,𝑝,𝑠,𝑥   𝐶,𝑚,𝑜,𝑝,𝑠,𝑥   𝐴,𝑐,𝑚,𝑜,𝑝,𝑠   𝑆,𝑐,𝑠,𝑥,𝑦   𝑇,𝑐,𝑚,𝑜,𝑝,𝑠,𝑥,𝑦   𝜑,𝑐,𝑚,𝑜,𝑝,𝑠,𝑥,𝑦   𝑉,𝑐,𝑥   𝐾,𝑐,𝑚,𝑜,𝑝,𝑠,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐶(𝑦,𝑐)   𝐷(𝑥,𝑦,𝑚,𝑜,𝑠,𝑝,𝑐)   𝑆(𝑚,𝑜,𝑝)   𝐸(𝑥,𝑦)   𝐻(𝑦)   𝑉(𝑦,𝑚,𝑜,𝑠,𝑝)

Proof of Theorem mclsssvlem

Step	Hyp	Ref	Expression
1		mclsval.3	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐸)
2		mclsval.1	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ mFS)
3		eqid 2734	. . . . . . 7 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
4		mclsval.e	. . . . . . 7 ⊢ 𝐸 = (mEx‘𝑇)
5		mclsval.h	. . . . . . 7 ⊢ 𝐻 = (mVH‘𝑇)
6	3, 4, 5	mvhf 35701	. . . . . 6 ⊢ (𝑇 ∈ mFS → 𝐻:(mVR‘𝑇)⟶𝐸)
7	2, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝐻:(mVR‘𝑇)⟶𝐸)
8	7	frnd 6668	. . . 4 ⊢ (𝜑 → ran 𝐻 ⊆ 𝐸)
9	1, 8	unssd 4142	. . 3 ⊢ (𝜑 → (𝐵 ∪ ran 𝐻) ⊆ 𝐸)
10		mclsval.s	. . . . . . . . . 10 ⊢ 𝑆 = (mSubst‘𝑇)
11	10, 4	msubf 35675	. . . . . . . . 9 ⊢ (𝑠 ∈ ran 𝑆 → 𝑠:𝐸⟶𝐸)
12		mclsval.a	. . . . . . . . . . . . . 14 ⊢ 𝐴 = (mAx‘𝑇)
13		eqid 2734	. . . . . . . . . . . . . 14 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
14	12, 13	maxsta 35697	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ mFS → 𝐴 ⊆ (mStat‘𝑇))
15	2, 14	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ (mStat‘𝑇))
16		eqid 2734	. . . . . . . . . . . . 13 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
17	16, 13	mstapst 35690	. . . . . . . . . . . 12 ⊢ (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
18	15, 17	sstrdi 3944	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ (mPreSt‘𝑇))
19	18	sselda 3931	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴) → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇))
20		mclsval.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (mDV‘𝑇)
21	20, 4, 16	elmpst 35679	. . . . . . . . . . 11 ⊢ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑚 ⊆ 𝐷 ∧ ◡𝑚 = 𝑚) ∧ (𝑜 ⊆ 𝐸 ∧ 𝑜 ∈ Fin) ∧ 𝑝 ∈ 𝐸))
22	21	simp3bi 1147	. . . . . . . . . 10 ⊢ (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇) → 𝑝 ∈ 𝐸)
23	19, 22	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴) → 𝑝 ∈ 𝐸)
24		ffvelcdm 7024	. . . . . . . . 9 ⊢ ((𝑠:𝐸⟶𝐸 ∧ 𝑝 ∈ 𝐸) → (𝑠‘𝑝) ∈ 𝐸)
25	11, 23, 24	syl2anr 597	. . . . . . . 8 ⊢ (((𝜑 ∧ ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴) ∧ 𝑠 ∈ ran 𝑆) → (𝑠‘𝑝) ∈ 𝐸)
26	25	a1d 25	. . . . . . 7 ⊢ (((𝜑 ∧ ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴) ∧ 𝑠 ∈ ran 𝑆) → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸))
27	26	ralrimiva 3126	. . . . . 6 ⊢ ((𝜑 ∧ ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴) → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸))
28	27	ex 412	. . . . 5 ⊢ (𝜑 → (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸)))
29	28	alrimiv 1928	. . . 4 ⊢ (𝜑 → ∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸)))
30	29	alrimivv 1929	. . 3 ⊢ (𝜑 → ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸)))
31	4	fvexi 6846	. . . 4 ⊢ 𝐸 ∈ V
32		sseq2 3958	. . . . 5 ⊢ (𝑐 = 𝐸 → ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ↔ (𝐵 ∪ ran 𝐻) ⊆ 𝐸))
33		sseq2 3958	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐸 → ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ↔ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸))
34	33	anbi1d 631	. . . . . . . . . 10 ⊢ (𝑐 = 𝐸 → (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) ↔ ((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾))))
35		eleq2 2823	. . . . . . . . . 10 ⊢ (𝑐 = 𝐸 → ((𝑠‘𝑝) ∈ 𝑐 ↔ (𝑠‘𝑝) ∈ 𝐸))
36	34, 35	imbi12d 344	. . . . . . . . 9 ⊢ (𝑐 = 𝐸 → ((((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐) ↔ (((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸)))
37	36	ralbidv 3157	. . . . . . . 8 ⊢ (𝑐 = 𝐸 → (∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐) ↔ ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸)))
38	37	imbi2d 340	. . . . . . 7 ⊢ (𝑐 = 𝐸 → ((⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) ↔ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸))))
39	38	albidv 1921	. . . . . 6 ⊢ (𝑐 = 𝐸 → (∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) ↔ ∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸))))
40	39	2albidv 1924	. . . . 5 ⊢ (𝑐 = 𝐸 → (∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) ↔ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸))))
41	32, 40	anbi12d 632	. . . 4 ⊢ (𝑐 = 𝐸 → (((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) ↔ ((𝐵 ∪ ran 𝐻) ⊆ 𝐸 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸)))))
42	31, 41	elab 3632	. . 3 ⊢ (𝐸 ∈ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ↔ ((𝐵 ∪ ran 𝐻) ⊆ 𝐸 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝐸 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝐸))))
43	9, 30, 42	sylanbrc 583	. 2 ⊢ (𝜑 → 𝐸 ∈ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))})
44		intss1 4916	. 2 ⊢ (𝐸 ∈ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} → ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ 𝐸)
45	43, 44	syl 17	1 ⊢ (𝜑 → ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ 𝐸)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2113  {cab 2712  ∀wral 3049   ∪ cun 3897   ⊆ wss 3899  ⟨cotp 4586  ∩ cint 4900   class class class wbr 5096   × cxp 5620  ^◡ccnv 5621  ran crn 5623   “ cima 5625  ⟶wf 6486  ‘cfv 6490  Fincfn 8881  mVRcmvar 35604  mAxcmax 35608  mExcmex 35610  mDVcmdv 35611  mVarscmvrs 35612  mSubstcmsub 35614  mVHcmvh 35615  mPreStcmpst 35616  mStatcmsta 35618  mFScmfs 35619  mClscmcls 35620

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 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

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 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-ot 4587  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-map 8763  df-pm 8764  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-n0 12400  df-z 12487  df-uz 12750  df-fz 13422  df-fzo 13569  df-seq 13923  df-hash 14252  df-word 14435  df-concat 14492  df-s1 14518  df-struct 17072  df-sets 17089  df-slot 17107  df-ndx 17119  df-base 17135  df-ress 17156  df-plusg 17188  df-0g 17359  df-gsum 17360  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-submnd 18707  df-frmd 18772  df-mrex 35629  df-mex 35630  df-mrsub 35633  df-msub 35634  df-mvh 35635  df-mpst 35636  df-msr 35637  df-msta 35638  df-mfs 35639

This theorem is referenced by:  mclsval  35706  mclsssv  35707