Theorem ss2mcls 33430

Description: The closure is monotonic under subsets of the original set of expressions and the set of disjoint variable conditions. (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)
ss2mcls.4	⊢ (𝜑 → 𝑋 ⊆ 𝐾)
ss2mcls.5	⊢ (𝜑 → 𝑌 ⊆ 𝐵)

Assertion

Ref	Expression
ss2mcls	⊢ (𝜑 → (𝑋𝐶𝑌) ⊆ (𝐾𝐶𝐵))

Proof of Theorem ss2mcls

Dummy variables 𝑐 𝑚 𝑜 𝑝 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ss2mcls.5	. . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ 𝐵)
2		unss1 4109	. . . . . 6 ⊢ (𝑌 ⊆ 𝐵 → (𝑌 ∪ ran (mVH‘𝑇)) ⊆ (𝐵 ∪ ran (mVH‘𝑇)))
3		sstr2 3924	. . . . . 6 ⊢ ((𝑌 ∪ ran (mVH‘𝑇)) ⊆ (𝐵 ∪ ran (mVH‘𝑇)) → ((𝐵 ∪ ran (mVH‘𝑇)) ⊆ 𝑐 → (𝑌 ∪ ran (mVH‘𝑇)) ⊆ 𝑐))
4	1, 2, 3	3syl 18	. . . . 5 ⊢ (𝜑 → ((𝐵 ∪ ran (mVH‘𝑇)) ⊆ 𝑐 → (𝑌 ∪ ran (mVH‘𝑇)) ⊆ 𝑐))
5		ss2mcls.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ⊆ 𝐾)
6		sstr2 3924	. . . . . . . . . . . . . 14 ⊢ ((((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝑋 → (𝑋 ⊆ 𝐾 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝐾))
7	5, 6	syl5com 31	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝑋 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝐾))
8	7	imim2d 57	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝑋) → (𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝐾)))
9	8	2alimdv 1922	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝑋) → ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝐾)))
10	9	anim2d 611	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝑋)) → ((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝐾))))
11	10	imim1d 82	. . . . . . . . 9 ⊢ (𝜑 → ((((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐) → (((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝑋)) → (𝑠‘𝑝) ∈ 𝑐)))
12	11	ralimdv 3103	. . . . . . . 8 ⊢ (𝜑 → (∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝑋)) → (𝑠‘𝑝) ∈ 𝑐)))
13	12	imim2d 57	. . . . . . 7 ⊢ (𝜑 → ((⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) → (⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝑋)) → (𝑠‘𝑝) ∈ 𝑐))))
14	13	alimdv 1920	. . . . . 6 ⊢ (𝜑 → (∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) → ∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝑋)) → (𝑠‘𝑝) ∈ 𝑐))))
15	14	2alimdv 1922	. . . . 5 ⊢ (𝜑 → (∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)) → ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝑋)) → (𝑠‘𝑝) ∈ 𝑐))))
16	4, 15	anim12d 608	. . . 4 ⊢ (𝜑 → (((𝐵 ∪ ran (mVH‘𝑇)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐))) → ((𝑌 ∪ ran (mVH‘𝑇)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝑋)) → (𝑠‘𝑝) ∈ 𝑐)))))
17	16	ss2abdv 3993	. . 3 ⊢ (𝜑 → {𝑐 ∣ ((𝐵 ∪ ran (mVH‘𝑇)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ {𝑐 ∣ ((𝑌 ∪ ran (mVH‘𝑇)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝑋)) → (𝑠‘𝑝) ∈ 𝑐)))})
18		intss 4897	. . 3 ⊢ ({𝑐 ∣ ((𝐵 ∪ ran (mVH‘𝑇)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ {𝑐 ∣ ((𝑌 ∪ ran (mVH‘𝑇)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝑋)) → (𝑠‘𝑝) ∈ 𝑐)))} → ∩ {𝑐 ∣ ((𝑌 ∪ ran (mVH‘𝑇)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝑋)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ ∩ {𝑐 ∣ ((𝐵 ∪ ran (mVH‘𝑇)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))})
19	17, 18	syl 17	. 2 ⊢ (𝜑 → ∩ {𝑐 ∣ ((𝑌 ∪ ran (mVH‘𝑇)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝑋)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ ∩ {𝑐 ∣ ((𝐵 ∪ ran (mVH‘𝑇)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))})
20		mclsval.d	. . 3 ⊢ 𝐷 = (mDV‘𝑇)
21		mclsval.e	. . 3 ⊢ 𝐸 = (mEx‘𝑇)
22		mclsval.c	. . 3 ⊢ 𝐶 = (mCls‘𝑇)
23		mclsval.1	. . 3 ⊢ (𝜑 → 𝑇 ∈ mFS)
24		mclsval.2	. . . 4 ⊢ (𝜑 → 𝐾 ⊆ 𝐷)
25	5, 24	sstrd 3927	. . 3 ⊢ (𝜑 → 𝑋 ⊆ 𝐷)
26		mclsval.3	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐸)
27	1, 26	sstrd 3927	. . 3 ⊢ (𝜑 → 𝑌 ⊆ 𝐸)
28		eqid 2738	. . 3 ⊢ (mVH‘𝑇) = (mVH‘𝑇)
29		eqid 2738	. . 3 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
30		eqid 2738	. . 3 ⊢ (mSubst‘𝑇) = (mSubst‘𝑇)
31		eqid 2738	. . 3 ⊢ (mVars‘𝑇) = (mVars‘𝑇)
32	20, 21, 22, 23, 25, 27, 28, 29, 30, 31	mclsval 33425	. 2 ⊢ (𝜑 → (𝑋𝐶𝑌) = ∩ {𝑐 ∣ ((𝑌 ∪ ran (mVH‘𝑇)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝑋)) → (𝑠‘𝑝) ∈ 𝑐)))})
33	20, 21, 22, 23, 24, 26, 28, 29, 30, 31	mclsval 33425	. 2 ⊢ (𝜑 → (𝐾𝐶𝐵) = ∩ {𝑐 ∣ ((𝐵 ∪ ran (mVH‘𝑇)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇) → ∀𝑠 ∈ ran (mSubst‘𝑇)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑇))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑥))) × ((mVars‘𝑇)‘(𝑠‘((mVH‘𝑇)‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))})
34	19, 32, 33	3sstr4d 3964	1 ⊢ (𝜑 → (𝑋𝐶𝑌) ⊆ (𝐾𝐶𝐵))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063   ∪ cun 3881   ⊆ wss 3883  ⟨cotp 4566  ∩ cint 4876   class class class wbr 5070   × cxp 5578  ran crn 5581   “ cima 5583  ‘cfv 6418  (class class class)co 7255  mAxcmax 33327  mExcmex 33329  mDVcmdv 33330  mVarscmvrs 33331  mSubstcmsub 33333  mVHcmvh 33334  mFScmfs 33338  mClscmcls 33339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-ot 4567  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-seq 13650  df-hash 13973  df-word 14146  df-concat 14202  df-s1 14229  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-0g 17069  df-gsum 17070  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-frmd 18403  df-mrex 33348  df-mex 33349  df-mrsub 33352  df-msub 33353  df-mvh 33354  df-mpst 33355  df-msr 33356  df-msta 33357  df-mfs 33358  df-mcls 33359

This theorem is referenced by:  mthmpps  33444