Theorem mthmpps 33107

Description: Given a theorem, there is an explicitly definable witnessing provable pre-statement for the provability of the theorem. (However, this pre-statement requires infinitely many disjoint variable conditions, which is sometimes inconvenient.) (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mthmpps.r	⊢ 𝑅 = (mStRed‘𝑇)
mthmpps.j	⊢ 𝐽 = (mPPSt‘𝑇)
mthmpps.u	⊢ 𝑈 = (mThm‘𝑇)
mthmpps.d	⊢ 𝐷 = (mDV‘𝑇)
mthmpps.v	⊢ 𝑉 = (mVars‘𝑇)
mthmpps.z	⊢ 𝑍 = ∪ (𝑉 “ (𝐻 ∪ {𝐴}))
mthmpps.m	⊢ 𝑀 = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍)))

Assertion

Ref	Expression
mthmpps	⊢ (𝑇 ∈ mFS → (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 ↔ (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))))

Proof of Theorem mthmpps

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mthmpps.m	. . . . . . . 8 ⊢ 𝑀 = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍)))
2		mthmpps.u	. . . . . . . . . . . . . 14 ⊢ 𝑈 = (mThm‘𝑇)
3		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
4	2, 3	mthmsta 33103	. . . . . . . . . . . . 13 ⊢ 𝑈 ⊆ (mPreSt‘𝑇)
5		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈)
6	4, 5	sseldi 3873	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇))
7		mthmpps.d	. . . . . . . . . . . . 13 ⊢ 𝐷 = (mDV‘𝑇)
8		eqid 2738	. . . . . . . . . . . . 13 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
9	7, 8, 3	elmpst 33061	. . . . . . . . . . . 12 ⊢ (⟨𝐶, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ ((𝐶 ⊆ 𝐷 ∧ ◡𝐶 = 𝐶) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
10	6, 9	sylib 221	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ((𝐶 ⊆ 𝐷 ∧ ◡𝐶 = 𝐶) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
11	10	simp1d 1143	. . . . . . . . . 10 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶 ⊆ 𝐷 ∧ ◡𝐶 = 𝐶))
12	11	simpld 498	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐶 ⊆ 𝐷)
13		difssd 4021	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐷 ∖ (𝑍 × 𝑍)) ⊆ 𝐷)
14	12, 13	unssd 4074	. . . . . . . 8 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ⊆ 𝐷)
15	1, 14	eqsstrid 3923	. . . . . . 7 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝑀 ⊆ 𝐷)
16	11	simprd 499	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ◡𝐶 = 𝐶)
17		cnvdif 5970	. . . . . . . . . . 11 ⊢ ◡(𝐷 ∖ (𝑍 × 𝑍)) = (◡𝐷 ∖ ◡(𝑍 × 𝑍))
18		cnvdif 5970	. . . . . . . . . . . . . 14 ⊢ ◡(((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (◡((mVR‘𝑇) × (mVR‘𝑇)) ∖ ◡ I )
19		cnvxp 5983	. . . . . . . . . . . . . . 15 ⊢ ◡((mVR‘𝑇) × (mVR‘𝑇)) = ((mVR‘𝑇) × (mVR‘𝑇))
20		cnvi 5968	. . . . . . . . . . . . . . 15 ⊢ ◡ I = I
21	19, 20	difeq12i 4009	. . . . . . . . . . . . . 14 ⊢ (◡((mVR‘𝑇) × (mVR‘𝑇)) ∖ ◡ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
22	18, 21	eqtri 2761	. . . . . . . . . . . . 13 ⊢ ◡(((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
23		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
24	23, 7	mdvval 33029	. . . . . . . . . . . . . 14 ⊢ 𝐷 = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
25	24	cnveqi 5711	. . . . . . . . . . . . 13 ⊢ ◡𝐷 = ◡(((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
26	22, 25, 24	3eqtr4i 2771	. . . . . . . . . . . 12 ⊢ ◡𝐷 = 𝐷
27		cnvxp 5983	. . . . . . . . . . . 12 ⊢ ◡(𝑍 × 𝑍) = (𝑍 × 𝑍)
28	26, 27	difeq12i 4009	. . . . . . . . . . 11 ⊢ (◡𝐷 ∖ ◡(𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍))
29	17, 28	eqtri 2761	. . . . . . . . . 10 ⊢ ◡(𝐷 ∖ (𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍))
30	29	a1i 11	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ◡(𝐷 ∖ (𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍)))
31	16, 30	uneq12d 4052	. . . . . . . 8 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (◡𝐶 ∪ ◡(𝐷 ∖ (𝑍 × 𝑍))) = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))))
32	1	cnveqi 5711	. . . . . . . . 9 ⊢ ◡𝑀 = ◡(𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍)))
33		cnvun 5969	. . . . . . . . 9 ⊢ ◡(𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) = (◡𝐶 ∪ ◡(𝐷 ∖ (𝑍 × 𝑍)))
34	32, 33	eqtri 2761	. . . . . . . 8 ⊢ ◡𝑀 = (◡𝐶 ∪ ◡(𝐷 ∖ (𝑍 × 𝑍)))
35	31, 34, 1	3eqtr4g 2798	. . . . . . 7 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ◡𝑀 = 𝑀)
36	15, 35	jca 515	. . . . . 6 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑀 ⊆ 𝐷 ∧ ◡𝑀 = 𝑀))
37	10	simp2d 1144	. . . . . 6 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin))
38	10	simp3d 1145	. . . . . 6 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐴 ∈ (mEx‘𝑇))
39	7, 8, 3	elmpst 33061	. . . . . 6 ⊢ (⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑀 ⊆ 𝐷 ∧ ◡𝑀 = 𝑀) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
40	36, 37, 38, 39	syl3anbrc 1344	. . . . 5 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇))
41		mthmpps.r	. . . . . . . 8 ⊢ 𝑅 = (mStRed‘𝑇)
42		mthmpps.j	. . . . . . . 8 ⊢ 𝐽 = (mPPSt‘𝑇)
43	41, 42, 2	elmthm 33101	. . . . . . 7 ⊢ (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
44	5, 43	sylib 221	. . . . . 6 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
45		eqid 2738	. . . . . . . 8 ⊢ (mCls‘𝑇) = (mCls‘𝑇)
46		simpll 767	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑇 ∈ mFS)
47	15	adantr 484	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑀 ⊆ 𝐷)
48	37	simpld 498	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐻 ⊆ (mEx‘𝑇))
49	48	adantr 484	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐻 ⊆ (mEx‘𝑇))
50	3, 42	mppspst 33099	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐽 ⊆ (mPreSt‘𝑇)
51		simprl 771	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 ∈ 𝐽)
52	50, 51	sseldi 3873	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 ∈ (mPreSt‘𝑇))
53	3	mpst123 33065	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (mPreSt‘𝑇) → 𝑥 = ⟨(1st ‘(1st ‘𝑥)), (2nd ‘(1st ‘𝑥)), (2nd ‘𝑥)⟩)
54	52, 53	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 = ⟨(1st ‘(1st ‘𝑥)), (2nd ‘(1st ‘𝑥)), (2nd ‘𝑥)⟩)
55	54	fveq2d 6672	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅‘𝑥) = (𝑅‘⟨(1st ‘(1st ‘𝑥)), (2nd ‘(1st ‘𝑥)), (2nd ‘𝑥)⟩))
56		simprr 773	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
57	55, 56	eqtr3d 2775	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅‘⟨(1st ‘(1st ‘𝑥)), (2nd ‘(1st ‘𝑥)), (2nd ‘𝑥)⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
58	54, 52	eqeltrrd 2834	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1st ‘(1st ‘𝑥)), (2nd ‘(1st ‘𝑥)), (2nd ‘𝑥)⟩ ∈ (mPreSt‘𝑇))
59		mthmpps.v	. . . . . . . . . . . . . . . . 17 ⊢ 𝑉 = (mVars‘𝑇)
60		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ ∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})) = ∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)}))
61	59, 3, 41, 60	msrval 33063	. . . . . . . . . . . . . . . 16 ⊢ (⟨(1st ‘(1st ‘𝑥)), (2nd ‘(1st ‘𝑥)), (2nd ‘𝑥)⟩ ∈ (mPreSt‘𝑇) → (𝑅‘⟨(1st ‘(1st ‘𝑥)), (2nd ‘(1st ‘𝑥)), (2nd ‘𝑥)⟩) = ⟨((1st ‘(1st ‘𝑥)) ∩ (∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})) × ∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})))), (2nd ‘(1st ‘𝑥)), (2nd ‘𝑥)⟩)
62	58, 61	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅‘⟨(1st ‘(1st ‘𝑥)), (2nd ‘(1st ‘𝑥)), (2nd ‘𝑥)⟩) = ⟨((1st ‘(1st ‘𝑥)) ∩ (∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})) × ∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})))), (2nd ‘(1st ‘𝑥)), (2nd ‘𝑥)⟩)
63		mthmpps.z	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑍 = ∪ (𝑉 “ (𝐻 ∪ {𝐴}))
64	59, 3, 41, 63	msrval 33063	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝐶, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → (𝑅‘⟨𝐶, 𝐻, 𝐴⟩) = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
65	6, 64	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑅‘⟨𝐶, 𝐻, 𝐴⟩) = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
66	65	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅‘⟨𝐶, 𝐻, 𝐴⟩) = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
67	57, 62, 66	3eqtr3d 2781	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨((1st ‘(1st ‘𝑥)) ∩ (∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})) × ∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})))), (2nd ‘(1st ‘𝑥)), (2nd ‘𝑥)⟩ = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
68		fvex 6681	. . . . . . . . . . . . . . . 16 ⊢ (1st ‘(1st ‘𝑥)) ∈ V
69	68	inex1 5182	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘(1st ‘𝑥)) ∩ (∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})) × ∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})))) ∈ V
70		fvex 6681	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘(1st ‘𝑥)) ∈ V
71		fvex 6681	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝑥) ∈ V
72	69, 70, 71	otth 5339	. . . . . . . . . . . . . 14 ⊢ (⟨((1st ‘(1st ‘𝑥)) ∩ (∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})) × ∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})))), (2nd ‘(1st ‘𝑥)), (2nd ‘𝑥)⟩ = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ ↔ (((1st ‘(1st ‘𝑥)) ∩ (∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})) × ∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})))) = (𝐶 ∩ (𝑍 × 𝑍)) ∧ (2nd ‘(1st ‘𝑥)) = 𝐻 ∧ (2nd ‘𝑥) = 𝐴))
73	67, 72	sylib 221	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (((1st ‘(1st ‘𝑥)) ∩ (∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})) × ∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})))) = (𝐶 ∩ (𝑍 × 𝑍)) ∧ (2nd ‘(1st ‘𝑥)) = 𝐻 ∧ (2nd ‘𝑥) = 𝐴))
74	73	simp1d 1143	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st ‘𝑥)) ∩ (∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})) × ∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})))) = (𝐶 ∩ (𝑍 × 𝑍)))
75	73	simp2d 1144	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (2nd ‘(1st ‘𝑥)) = 𝐻)
76	73	simp3d 1145	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (2nd ‘𝑥) = 𝐴)
77	76	sneqd 4525	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → {(2nd ‘𝑥)} = {𝐴})
78	75, 77	uneq12d 4052	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)}) = (𝐻 ∪ {𝐴}))
79	78	imaeq2d 5897	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})) = (𝑉 “ (𝐻 ∪ {𝐴})))
80	79	unieqd 4807	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})) = ∪ (𝑉 “ (𝐻 ∪ {𝐴})))
81	80, 63	eqtr4di 2791	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})) = 𝑍)
82	81	sqxpeqd 5551	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})) × ∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)}))) = (𝑍 × 𝑍))
83	82	ineq2d 4101	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st ‘𝑥)) ∩ (∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})) × ∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})))) = ((1st ‘(1st ‘𝑥)) ∩ (𝑍 × 𝑍)))
84	74, 83	eqtr3d 2775	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝐶 ∩ (𝑍 × 𝑍)) = ((1st ‘(1st ‘𝑥)) ∩ (𝑍 × 𝑍)))
85		inss1 4117	. . . . . . . . . . 11 ⊢ (𝐶 ∩ (𝑍 × 𝑍)) ⊆ 𝐶
86	84, 85	eqsstrrdi 3930	. . . . . . . . . 10 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st ‘𝑥)) ∩ (𝑍 × 𝑍)) ⊆ 𝐶)
87		eqidd 2739	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑥)))
88	87, 75, 76	oteq123d 4773	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1st ‘(1st ‘𝑥)), (2nd ‘(1st ‘𝑥)), (2nd ‘𝑥)⟩ = ⟨(1st ‘(1st ‘𝑥)), 𝐻, 𝐴⟩)
89	54, 88	eqtrd 2773	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 = ⟨(1st ‘(1st ‘𝑥)), 𝐻, 𝐴⟩)
90	89, 52	eqeltrrd 2834	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1st ‘(1st ‘𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇))
91	7, 8, 3	elmpst 33061	. . . . . . . . . . . . . 14 ⊢ (⟨(1st ‘(1st ‘𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ (((1st ‘(1st ‘𝑥)) ⊆ 𝐷 ∧ ◡(1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑥))) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
92	91	simp1bi 1146	. . . . . . . . . . . . 13 ⊢ (⟨(1st ‘(1st ‘𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → ((1st ‘(1st ‘𝑥)) ⊆ 𝐷 ∧ ◡(1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑥))))
93	92	simpld 498	. . . . . . . . . . . 12 ⊢ (⟨(1st ‘(1st ‘𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → (1st ‘(1st ‘𝑥)) ⊆ 𝐷)
94	90, 93	syl 17	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1st ‘(1st ‘𝑥)) ⊆ 𝐷)
95	94	ssdifd 4029	. . . . . . . . . 10 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st ‘𝑥)) ∖ (𝑍 × 𝑍)) ⊆ (𝐷 ∖ (𝑍 × 𝑍)))
96		unss12 4070	. . . . . . . . . 10 ⊢ ((((1st ‘(1st ‘𝑥)) ∩ (𝑍 × 𝑍)) ⊆ 𝐶 ∧ ((1st ‘(1st ‘𝑥)) ∖ (𝑍 × 𝑍)) ⊆ (𝐷 ∖ (𝑍 × 𝑍))) → (((1st ‘(1st ‘𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1st ‘(1st ‘𝑥)) ∖ (𝑍 × 𝑍))) ⊆ (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))))
97	86, 95, 96	syl2anc 587	. . . . . . . . 9 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (((1st ‘(1st ‘𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1st ‘(1st ‘𝑥)) ∖ (𝑍 × 𝑍))) ⊆ (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))))
98		inundif 4365	. . . . . . . . . 10 ⊢ (((1st ‘(1st ‘𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1st ‘(1st ‘𝑥)) ∖ (𝑍 × 𝑍))) = (1st ‘(1st ‘𝑥))
99	98	eqcomi 2747	. . . . . . . . 9 ⊢ (1st ‘(1st ‘𝑥)) = (((1st ‘(1st ‘𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1st ‘(1st ‘𝑥)) ∖ (𝑍 × 𝑍)))
100	97, 99, 1	3sstr4g 3920	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1st ‘(1st ‘𝑥)) ⊆ 𝑀)
101		ssidd 3898	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐻 ⊆ 𝐻)
102	7, 8, 45, 46, 47, 49, 100, 101	ss2mcls 33093	. . . . . . 7 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st ‘𝑥))(mCls‘𝑇)𝐻) ⊆ (𝑀(mCls‘𝑇)𝐻))
103	89, 51	eqeltrrd 2834	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1st ‘(1st ‘𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽)
104	3, 42, 45	elmpps 33098	. . . . . . . . 9 ⊢ (⟨(1st ‘(1st ‘𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨(1st ‘(1st ‘𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ∧ 𝐴 ∈ ((1st ‘(1st ‘𝑥))(mCls‘𝑇)𝐻)))
105	104	simprbi 500	. . . . . . . 8 ⊢ (⟨(1st ‘(1st ‘𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽 → 𝐴 ∈ ((1st ‘(1st ‘𝑥))(mCls‘𝑇)𝐻))
106	103, 105	syl 17	. . . . . . 7 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐴 ∈ ((1st ‘(1st ‘𝑥))(mCls‘𝑇)𝐻))
107	102, 106	sseldd 3876	. . . . . 6 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻))
108	44, 107	rexlimddv 3200	. . . . 5 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻))
109	3, 42, 45	elmpps 33098	. . . . 5 ⊢ (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ∧ 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻)))
110	40, 108, 109	sylanbrc 586	. . . 4 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽)
111	1	ineq1i 4097	. . . . . . . 8 ⊢ (𝑀 ∩ (𝑍 × 𝑍)) = ((𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ∩ (𝑍 × 𝑍))
112		indir 4164	. . . . . . . 8 ⊢ ((𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ∩ (𝑍 × 𝑍)) = ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)))
113		disjdifr 4359	. . . . . . . . . 10 ⊢ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) = ∅
114		0ss 4282	. . . . . . . . . 10 ⊢ ∅ ⊆ (𝐶 ∩ (𝑍 × 𝑍))
115	113, 114	eqsstri 3909	. . . . . . . . 9 ⊢ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) ⊆ (𝐶 ∩ (𝑍 × 𝑍))
116		ssequn2 4071	. . . . . . . . 9 ⊢ (((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) ⊆ (𝐶 ∩ (𝑍 × 𝑍)) ↔ ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍))) = (𝐶 ∩ (𝑍 × 𝑍)))
117	115, 116	mpbi 233	. . . . . . . 8 ⊢ ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍))) = (𝐶 ∩ (𝑍 × 𝑍))
118	111, 112, 117	3eqtri 2765	. . . . . . 7 ⊢ (𝑀 ∩ (𝑍 × 𝑍)) = (𝐶 ∩ (𝑍 × 𝑍))
119	118	a1i 11	. . . . . 6 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑀 ∩ (𝑍 × 𝑍)) = (𝐶 ∩ (𝑍 × 𝑍)))
120	119	oteq1d 4770	. . . . 5 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨(𝑀 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
121	59, 3, 41, 63	msrval 33063	. . . . . 6 ⊢ (⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = ⟨(𝑀 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
122	40, 121	syl 17	. . . . 5 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = ⟨(𝑀 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
123	120, 122, 65	3eqtr4d 2783	. . . 4 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
124	110, 123	jca 515	. . 3 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩)))
125	124	ex 416	. 2 ⊢ (𝑇 ∈ mFS → (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 → (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))))
126	41, 42, 2	mthmi 33102	. 2 ⊢ ((⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩)) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈)
127	125, 126	impbid1 228	1 ⊢ (𝑇 ∈ mFS → (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 ↔ (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2113  ∃wrex 3054   ∖ cdif 3838   ∪ cun 3839   ∩ cin 3840   ⊆ wss 3841  ∅c0 4209  {csn 4513  ⟨cotp 4521  ∪ cuni 4793   I cid 5424   × cxp 5517  ^◡ccnv 5518   “ cima 5522  ‘cfv 6333  (class class class)co 7164  1^st c1st 7705  2^nd c2nd 7706  Fincfn 8548  mVRcmvar 32986  mExcmex 32992  mDVcmdv 32993  mVarscmvrs 32994  mPreStcmpst 32998  mStRedcmsr 32999  mFScmfs 33001  mClscmcls 33002  mPPStcmpps 33003  mThmcmthm 33004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473  ax-cnex 10664  ax-resscn 10665  ax-1cn 10666  ax-icn 10667  ax-addcl 10668  ax-addrcl 10669  ax-mulcl 10670  ax-mulrcl 10671  ax-mulcom 10672  ax-addass 10673  ax-mulass 10674  ax-distr 10675  ax-i2m1 10676  ax-1ne0 10677  ax-1rid 10678  ax-rnegex 10679  ax-rrecex 10680  ax-cnre 10681  ax-pre-lttri 10682  ax-pre-lttrn 10683  ax-pre-ltadd 10684  ax-pre-mulgt0 10685

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-ot 4522  df-uni 4794  df-int 4834  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-1st 7707  df-2nd 7708  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-1o 8124  df-er 8313  df-map 8432  df-pm 8433  df-en 8549  df-dom 8550  df-sdom 8551  df-fin 8552  df-card 9434  df-pnf 10748  df-mnf 10749  df-xr 10750  df-ltxr 10751  df-le 10752  df-sub 10943  df-neg 10944  df-nn 11710  df-2 11772  df-n0 11970  df-z 12056  df-uz 12318  df-fz 12975  df-fzo 13118  df-seq 13454  df-hash 13776  df-word 13949  df-concat 14005  df-s1 14032  df-struct 16581  df-ndx 16582  df-slot 16583  df-base 16585  df-sets 16586  df-ress 16587  df-plusg 16674  df-0g 16811  df-gsum 16812  df-mgm 17961  df-sgrp 18010  df-mnd 18021  df-submnd 18066  df-frmd 18123  df-mrex 33011  df-mex 33012  df-mdv 33013  df-mrsub 33015  df-msub 33016  df-mvh 33017  df-mpst 33018  df-msr 33019  df-msta 33020  df-mfs 33021  df-mcls 33022  df-mpps 33023  df-mthm 33024

This theorem is referenced by: (None)