Theorem ss2mcls 35540

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 4138	. . . . . 6 ⊢ (𝑌 ⊆ 𝐵 → (𝑌 ∪ ran (mVH‘𝑇)) ⊆ (𝐵 ∪ ran (mVH‘𝑇)))
3		sstr2 3944	. . . . . 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 3944	. . . . . . . . . . . . . 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 1918	. . . . . . . . . . 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 612	. . . . . . . . . 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 3143	. . . . . . . 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 1916	. . . . . 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 1918	. . . . 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 609	. . . 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 4020	. . 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 4922	. . 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 3948	. . 3 ⊢ (𝜑 → 𝑋 ⊆ 𝐷)
26		mclsval.3	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐸)
27	1, 26	sstrd 3948	. . 3 ⊢ (𝜑 → 𝑌 ⊆ 𝐸)
28		eqid 2729	. . 3 ⊢ (mVH‘𝑇) = (mVH‘𝑇)
29		eqid 2729	. . 3 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
30		eqid 2729	. . 3 ⊢ (mSubst‘𝑇) = (mSubst‘𝑇)
31		eqid 2729	. . 3 ⊢ (mVars‘𝑇) = (mVars‘𝑇)
32	20, 21, 22, 23, 25, 27, 28, 29, 30, 31	mclsval 35535	. 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 35535	. 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 3993	1 ⊢ (𝜑 → (𝑋𝐶𝑌) ⊆ (𝐾𝐶𝐵))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044   ∪ cun 3903   ⊆ wss 3905  ⟨cotp 4587  ∩ cint 4899   class class class wbr 5095   × cxp 5621  ran crn 5624   “ cima 5626  ‘cfv 6486  (class class class)co 7353  mAxcmax 35437  mExcmex 35439  mDVcmdv 35440  mVarscmvrs 35441  mSubstcmsub 35443  mVHcmvh 35444  mFScmfs 35448  mClscmcls 35449

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 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

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 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-ot 4588  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  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 8632  df-map 8762  df-pm 8763  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-n0 12403  df-z 12490  df-uz 12754  df-fz 13429  df-fzo 13576  df-seq 13927  df-hash 14256  df-word 14439  df-concat 14496  df-s1 14521  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-0g 17363  df-gsum 17364  df-mgm 18532  df-sgrp 18611  df-mnd 18627  df-submnd 18676  df-frmd 18741  df-mrex 35458  df-mex 35459  df-mrsub 35462  df-msub 35463  df-mvh 35464  df-mpst 35465  df-msr 35466  df-msta 35467  df-mfs 35468  df-mcls 35469

This theorem is referenced by:  mthmpps  35554