Theorem ss2mcls 35211

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 4181	. . . . . 6 ⊢ (𝑌 ⊆ 𝐵 → (𝑌 ∪ ran (mVH‘𝑇)) ⊆ (𝐵 ∪ ran (mVH‘𝑇)))
3		sstr2 3989	. . . . . 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 3989	. . . . . . . . . . . . . 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 1913	. . . . . . . . . . 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 610	. . . . . . . . . 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 3166	. . . . . . . 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 1911	. . . . . 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 1913	. . . . 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 607	. . . 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 4060	. . 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 4976	. . 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 3992	. . 3 ⊢ (𝜑 → 𝑋 ⊆ 𝐷)
26		mclsval.3	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐸)
27	1, 26	sstrd 3992	. . 3 ⊢ (𝜑 → 𝑌 ⊆ 𝐸)
28		eqid 2728	. . 3 ⊢ (mVH‘𝑇) = (mVH‘𝑇)
29		eqid 2728	. . 3 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
30		eqid 2728	. . 3 ⊢ (mSubst‘𝑇) = (mSubst‘𝑇)
31		eqid 2728	. . 3 ⊢ (mVars‘𝑇) = (mVars‘𝑇)
32	20, 21, 22, 23, 25, 27, 28, 29, 30, 31	mclsval 35206	. 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 35206	. 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 4029	1 ⊢ (𝜑 → (𝑋𝐶𝑌) ⊆ (𝐾𝐶𝐵))

Syntax hints:   → wi 4   ∧ wa 394  ∀wal 1531   = wceq 1533   ∈ wcel 2098  {cab 2705  ∀wral 3058   ∪ cun 3947   ⊆ wss 3949  ⟨cotp 4640  ∩ cint 4953   class class class wbr 5152   × cxp 5680  ran crn 5683   “ cima 5685  ‘cfv 6553  (class class class)co 7426  mAxcmax 35108  mExcmex 35110  mDVcmdv 35111  mVarscmvrs 35112  mSubstcmsub 35114  mVHcmvh 35115  mFScmfs 35119  mClscmcls 35120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-ot 4641  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-pm 8854  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-n0 12511  df-z 12597  df-uz 12861  df-fz 13525  df-fzo 13668  df-seq 14007  df-hash 14330  df-word 14505  df-concat 14561  df-s1 14586  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-0g 17430  df-gsum 17431  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-submnd 18748  df-frmd 18808  df-mrex 35129  df-mex 35130  df-mrsub 35133  df-msub 35134  df-mvh 35135  df-mpst 35136  df-msr 35137  df-msta 35138  df-mfs 35139  df-mcls 35140

This theorem is referenced by:  mthmpps  35225