Theorem ss2mcls 36302

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 4131	. . . . . 6 ⊢ (𝑌 ⊆ 𝐵 → (𝑌 ∪ ran (mVH‘𝑇)) ⊆ (𝐵 ∪ ran (mVH‘𝑇)))
3		sstr2 3938	. . . . . 6 ⊢ ((𝑌 ∪ ran (mVH‘𝑇)) ⊆ (𝐵 ∪ ran (mVH‘𝑇)) → ((𝐵 ∪ ran (mVH‘𝑇)) ⊆ 𝑐 → (𝑌 ∪ ran (mVH‘𝑇)) ⊆ 𝑐))
4	1, 2, 3	3syl 19	. . . . 5 ⊢ (𝜑 → ((𝐵 ∪ ran (mVH‘𝑇)) ⊆ 𝑐 → (𝑌 ∪ ran (mVH‘𝑇)) ⊆ 𝑐))
5		ss2mcls.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ⊆ 𝐾)
6		sstr2 3938	. . . . . . . . . . . . . 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 32	. . . . . . . . . . . . 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 58	. . . . . . . . . . . 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 1951	. . . . . . . . . . 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 624	. . . . . . . . . 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 83	. . . . . . . . 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 3177	. . . . . . . 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 58	. . . . . . 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 1949	. . . . . 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 1951	. . . . 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 621	. . . 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 4013	. . 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 4929	. . 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 18	. 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 3941	. . 3 ⊢ (𝜑 → 𝑋 ⊆ 𝐷)
26		mclsval.3	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐸)
27	1, 26	sstrd 3941	. . 3 ⊢ (𝜑 → 𝑌 ⊆ 𝐸)
28		eqid 2761	. . 3 ⊢ (mVH‘𝑇) = (mVH‘𝑇)
29		eqid 2761	. . 3 ⊢ (mAx‘𝑇) = (mAx‘𝑇)
30		eqid 2761	. . 3 ⊢ (mSubst‘𝑇) = (mSubst‘𝑇)
31		eqid 2761	. . 3 ⊢ (mVars‘𝑇) = (mVars‘𝑇)
32	20, 21, 22, 23, 25, 27, 28, 29, 30, 31	mclsval 36297	. 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 36297	. 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 3986	1 ⊢ (𝜑 → (𝑋𝐶𝑌) ⊆ (𝐾𝐶𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077   ∪ cun 3897   ⊆ wss 3899  ⟨cotp 4592  ∩ cint 4907   class class class wbr 5103   × cxp 5649  ran crn 5652   “ cima 5654  ‘cfv 6531  (class class class)co 7412  mAxcmax 36199  mExcmex 36201  mDVcmdv 36202  mVarscmvrs 36203  mSubstcmsub 36205  mVHcmvh 36206  mFScmfs 36210  mClscmcls 36211

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-er 8701  df-map 8833  df-pm 8834  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-card 10001  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-2 12386  df-n0 12588  df-z 12675  df-uz 12947  df-fz 13621  df-fzo 13769  df-seq 14125  df-hash 14455  df-word 14639  df-concat 14696  df-s1 14723  df-struct 17305  df-sets 17322  df-slot 17340  df-ndx 17352  df-base 17368  df-ress 17389  df-plusg 17421  df-0g 17592  df-gsum 17593  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-submnd 18959  df-frmd 19025  df-mrex 36220  df-mex 36221  df-mrsub 36224  df-msub 36225  df-mvh 36226  df-mpst 36227  df-msr 36228  df-msta 36229  df-mfs 36230  df-mcls 36231

This theorem is used by:  mthmpps  36316