Theorem mclsssvlem 35549

Description: Lemma for mclsssv 35551. (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 2729	. . . . . . 7 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
4		mclsval.e	. . . . . . 7 ⊢ 𝐸 = (mEx‘𝑇)
5		mclsval.h	. . . . . . 7 ⊢ 𝐻 = (mVH‘𝑇)
6	3, 4, 5	mvhf 35545	. . . . . 6 ⊢ (𝑇 ∈ mFS → 𝐻:(mVR‘𝑇)⟶𝐸)
7	2, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝐻:(mVR‘𝑇)⟶𝐸)
8	7	frnd 6696	. . . 4 ⊢ (𝜑 → ran 𝐻 ⊆ 𝐸)
9	1, 8	unssd 4155	. . 3 ⊢ (𝜑 → (𝐵 ∪ ran 𝐻) ⊆ 𝐸)
10		mclsval.s	. . . . . . . . . 10 ⊢ 𝑆 = (mSubst‘𝑇)
11	10, 4	msubf 35519	. . . . . . . . 9 ⊢ (𝑠 ∈ ran 𝑆 → 𝑠:𝐸⟶𝐸)
12		mclsval.a	. . . . . . . . . . . . . 14 ⊢ 𝐴 = (mAx‘𝑇)
13		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (mStat‘𝑇) = (mStat‘𝑇)
14	12, 13	maxsta 35541	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ mFS → 𝐴 ⊆ (mStat‘𝑇))
15	2, 14	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ (mStat‘𝑇))
16		eqid 2729	. . . . . . . . . . . . 13 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
17	16, 13	mstapst 35534	. . . . . . . . . . . 12 ⊢ (mStat‘𝑇) ⊆ (mPreSt‘𝑇)
18	15, 17	sstrdi 3959	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ (mPreSt‘𝑇))
19	18	sselda 3946	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴) → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mPreSt‘𝑇))
20		mclsval.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (mDV‘𝑇)
21	20, 4, 16	elmpst 35523	. . . . . . . . . . 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 7053	. . . . . . . . 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 3125	. . . . . 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 1927	. . . 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 1928	. . 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 6872	. . . 4 ⊢ 𝐸 ∈ V
32		sseq2 3973	. . . . 5 ⊢ (𝑐 = 𝐸 → ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ↔ (𝐵 ∪ ran 𝐻) ⊆ 𝐸))
33		sseq2 3973	. . . . . . . . . . 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 2817	. . . . . . . . . 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 3156	. . . . . . . 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 1920	. . . . . 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 1923	. . . . 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 3646	. . 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 4927	. 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 1538   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044   ∪ cun 3912   ⊆ wss 3914  ⟨cotp 4597  ∩ cint 4910   class class class wbr 5107   × cxp 5636  ^◡ccnv 5637  ran crn 5639   “ cima 5641  ⟶wf 6507  ‘cfv 6511  Fincfn 8918  mVRcmvar 35448  mAxcmax 35452  mExcmex 35454  mDVcmdv 35455  mVarscmvrs 35456  mSubstcmsub 35458  mVHcmvh 35459  mPreStcmpst 35460  mStatcmsta 35462  mFScmfs 35463  mClscmcls 35464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-ot 4598  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-pm 8802  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-n0 12443  df-z 12530  df-uz 12794  df-fz 13469  df-fzo 13616  df-seq 13967  df-hash 14296  df-word 14479  df-concat 14536  df-s1 14561  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-0g 17404  df-gsum 17405  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-submnd 18711  df-frmd 18776  df-mrex 35473  df-mex 35474  df-mrsub 35477  df-msub 35478  df-mvh 35479  df-mpst 35480  df-msr 35481  df-msta 35482  df-mfs 35483

This theorem is referenced by:  mclsval  35550  mclsssv  35551