Theorem mthmpps 35609

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 2736	. . . . . . . . . . . . . 14 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
4	2, 3	mthmsta 35605	. . . . . . . . . . . . 13 ⊢ 𝑈 ⊆ (mPreSt‘𝑇)
5		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈)
6	4, 5	sselid 3961	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇))
7		mthmpps.d	. . . . . . . . . . . . 13 ⊢ 𝐷 = (mDV‘𝑇)
8		eqid 2736	. . . . . . . . . . . . 13 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
9	7, 8, 3	elmpst 35563	. . . . . . . . . . . 12 ⊢ (⟨𝐶, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ ((𝐶 ⊆ 𝐷 ∧ ◡𝐶 = 𝐶) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
10	6, 9	sylib 218	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ((𝐶 ⊆ 𝐷 ∧ ◡𝐶 = 𝐶) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
11	10	simp1d 1142	. . . . . . . . . 10 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶 ⊆ 𝐷 ∧ ◡𝐶 = 𝐶))
12	11	simpld 494	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐶 ⊆ 𝐷)
13		difssd 4117	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐷 ∖ (𝑍 × 𝑍)) ⊆ 𝐷)
14	12, 13	unssd 4172	. . . . . . . 8 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ⊆ 𝐷)
15	1, 14	eqsstrid 4002	. . . . . . 7 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝑀 ⊆ 𝐷)
16	11	simprd 495	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ◡𝐶 = 𝐶)
17		cnvdif 6137	. . . . . . . . . . 11 ⊢ ◡(𝐷 ∖ (𝑍 × 𝑍)) = (◡𝐷 ∖ ◡(𝑍 × 𝑍))
18		cnvdif 6137	. . . . . . . . . . . . . 14 ⊢ ◡(((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (◡((mVR‘𝑇) × (mVR‘𝑇)) ∖ ◡ I )
19		cnvxp 6151	. . . . . . . . . . . . . . 15 ⊢ ◡((mVR‘𝑇) × (mVR‘𝑇)) = ((mVR‘𝑇) × (mVR‘𝑇))
20		cnvi 6135	. . . . . . . . . . . . . . 15 ⊢ ◡ I = I
21	19, 20	difeq12i 4104	. . . . . . . . . . . . . 14 ⊢ (◡((mVR‘𝑇) × (mVR‘𝑇)) ∖ ◡ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
22	18, 21	eqtri 2759	. . . . . . . . . . . . 13 ⊢ ◡(((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
23		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
24	23, 7	mdvval 35531	. . . . . . . . . . . . . 14 ⊢ 𝐷 = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
25	24	cnveqi 5859	. . . . . . . . . . . . 13 ⊢ ◡𝐷 = ◡(((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
26	22, 25, 24	3eqtr4i 2769	. . . . . . . . . . . 12 ⊢ ◡𝐷 = 𝐷
27		cnvxp 6151	. . . . . . . . . . . 12 ⊢ ◡(𝑍 × 𝑍) = (𝑍 × 𝑍)
28	26, 27	difeq12i 4104	. . . . . . . . . . 11 ⊢ (◡𝐷 ∖ ◡(𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍))
29	17, 28	eqtri 2759	. . . . . . . . . 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 4149	. . . . . . . 8 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (◡𝐶 ∪ ◡(𝐷 ∖ (𝑍 × 𝑍))) = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))))
32	1	cnveqi 5859	. . . . . . . . 9 ⊢ ◡𝑀 = ◡(𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍)))
33		cnvun 6136	. . . . . . . . 9 ⊢ ◡(𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) = (◡𝐶 ∪ ◡(𝐷 ∖ (𝑍 × 𝑍)))
34	32, 33	eqtri 2759	. . . . . . . 8 ⊢ ◡𝑀 = (◡𝐶 ∪ ◡(𝐷 ∖ (𝑍 × 𝑍)))
35	31, 34, 1	3eqtr4g 2796	. . . . . . 7 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ◡𝑀 = 𝑀)
36	15, 35	jca 511	. . . . . 6 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑀 ⊆ 𝐷 ∧ ◡𝑀 = 𝑀))
37	10	simp2d 1143	. . . . . 6 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin))
38	10	simp3d 1144	. . . . . 6 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐴 ∈ (mEx‘𝑇))
39	7, 8, 3	elmpst 35563	. . . . . 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 35603	. . . . . . 7 ⊢ (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
44	5, 43	sylib 218	. . . . . 6 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
45		eqid 2736	. . . . . . . 8 ⊢ (mCls‘𝑇) = (mCls‘𝑇)
46		simpll 766	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑇 ∈ mFS)
47	15	adantr 480	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑀 ⊆ 𝐷)
48	37	simpld 494	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐻 ⊆ (mEx‘𝑇))
49	48	adantr 480	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐻 ⊆ (mEx‘𝑇))
50	3, 42	mppspst 35601	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐽 ⊆ (mPreSt‘𝑇)
51		simprl 770	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 ∈ 𝐽)
52	50, 51	sselid 3961	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 ∈ (mPreSt‘𝑇))
53	3	mpst123 35567	. . . . . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . 17 ⊢ ∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})) = ∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)}))
61	59, 3, 41, 60	msrval 35565	. . . . . . . . . . . . . . . 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 35565	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . 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 6894	. . . . . . . . . . . . . . . 16 ⊢ (1st ‘(1st ‘𝑥)) ∈ V
69	68	inex1 5292	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘(1st ‘𝑥)) ∩ (∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})) × ∪ (𝑉 “ ((2nd ‘(1st ‘𝑥)) ∪ {(2nd ‘𝑥)})))) ∈ V
70		fvex 6894	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘(1st ‘𝑥)) ∈ V
71		fvex 6894	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝑥) ∈ V
72	69, 70, 71	otth 5464	. . . . . . . . . . . . . 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 218	. . . . . . . . . . . . 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 1142	. . . . . . . . . . . 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 1143	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (2nd ‘(1st ‘𝑥)) = 𝐻)
76	73	simp3d 1144	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (2nd ‘𝑥) = 𝐴)
77	76	sneqd 4618	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → {(2nd ‘𝑥)} = {𝐴})
78	75, 77	uneq12d 4149	. . . . . . . . . . . . . . . . 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 6052	. . . . . . . . . . . . . . . 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 4901	. . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . 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 5691	. . . . . . . . . . . . 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 4200	. . . . . . . . . . . 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 2773	. . . . . . . . . . 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 4217	. . . . . . . . . . 11 ⊢ (𝐶 ∩ (𝑍 × 𝑍)) ⊆ 𝐶
86	84, 85	eqsstrrdi 4009	. . . . . . . . . 10 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1st ‘(1st ‘𝑥)) ∩ (𝑍 × 𝑍)) ⊆ 𝐶)
87		eqidd 2737	. . . . . . . . . . . . . . 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 4869	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . 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 2836	. . . . . . . . . . . 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 35563	. . . . . . . . . . . . . 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 1145	. . . . . . . . . . . . 13 ⊢ (⟨(1st ‘(1st ‘𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → ((1st ‘(1st ‘𝑥)) ⊆ 𝐷 ∧ ◡(1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑥))))
93	92	simpld 494	. . . . . . . . . . . 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 4125	. . . . . . . . . 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 4168	. . . . . . . . . 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 584	. . . . . . . . 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 4459	. . . . . . . . . 10 ⊢ (((1st ‘(1st ‘𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1st ‘(1st ‘𝑥)) ∖ (𝑍 × 𝑍))) = (1st ‘(1st ‘𝑥))
99	98	eqcomi 2745	. . . . . . . . 9 ⊢ (1st ‘(1st ‘𝑥)) = (((1st ‘(1st ‘𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1st ‘(1st ‘𝑥)) ∖ (𝑍 × 𝑍)))
100	97, 99, 1	3sstr4g 4017	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1st ‘(1st ‘𝑥)) ⊆ 𝑀)
101		ssidd 3987	. . . . . . . 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 35595	. . . . . . 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 2836	. . . . . . . 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 35600	. . . . . . . . 9 ⊢ (⟨(1st ‘(1st ‘𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨(1st ‘(1st ‘𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ∧ 𝐴 ∈ ((1st ‘(1st ‘𝑥))(mCls‘𝑇)𝐻)))
105	104	simprbi 496	. . . . . . . 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 3964	. . . . . 6 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻))
108	44, 107	rexlimddv 3148	. . . . 5 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻))
109	3, 42, 45	elmpps 35600	. . . . 5 ⊢ (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ∧ 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻)))
110	40, 108, 109	sylanbrc 583	. . . 4 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽)
111	1	ineq1i 4196	. . . . . . . 8 ⊢ (𝑀 ∩ (𝑍 × 𝑍)) = ((𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ∩ (𝑍 × 𝑍))
112		indir 4266	. . . . . . . 8 ⊢ ((𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ∩ (𝑍 × 𝑍)) = ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)))
113		disjdifr 4453	. . . . . . . . . 10 ⊢ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) = ∅
114		0ss 4380	. . . . . . . . . 10 ⊢ ∅ ⊆ (𝐶 ∩ (𝑍 × 𝑍))
115	113, 114	eqsstri 4010	. . . . . . . . 9 ⊢ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) ⊆ (𝐶 ∩ (𝑍 × 𝑍))
116		ssequn2 4169	. . . . . . . . 9 ⊢ (((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) ⊆ (𝐶 ∩ (𝑍 × 𝑍)) ↔ ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍))) = (𝐶 ∩ (𝑍 × 𝑍)))
117	115, 116	mpbi 230	. . . . . . . 8 ⊢ ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍))) = (𝐶 ∩ (𝑍 × 𝑍))
118	111, 112, 117	3eqtri 2763	. . . . . . 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 4866	. . . . 5 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨(𝑀 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
121	59, 3, 41, 63	msrval 35565	. . . . . 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 2781	. . . 4 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
124	110, 123	jca 511	. . 3 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩)))
125	124	ex 412	. 2 ⊢ (𝑇 ∈ mFS → (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 → (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))))
126	41, 42, 2	mthmi 35604	. 2 ⊢ ((⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩)) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈)
127	125, 126	impbid1 225	1 ⊢ (𝑇 ∈ mFS → (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 ↔ (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3061   ∖ cdif 3928   ∪ cun 3929   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313  {csn 4606  ⟨cotp 4614  ∪ cuni 4888   I cid 5552   × cxp 5657  ^◡ccnv 5658   “ cima 5662  ‘cfv 6536  (class class class)co 7410  1^st c1st 7991  2^nd c2nd 7992  Fincfn 8964  mVRcmvar 35488  mExcmex 35494  mDVcmdv 35495  mVarscmvrs 35496  mPreStcmpst 35500  mStRedcmsr 35501  mFScmfs 35503  mClscmcls 35504  mPPStcmpps 35505  mThmcmthm 35506

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-ot 4615  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-card 9958  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-n0 12507  df-z 12594  df-uz 12858  df-fz 13530  df-fzo 13677  df-seq 14025  df-hash 14354  df-word 14537  df-concat 14594  df-s1 14619  df-struct 17171  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-ress 17257  df-plusg 17289  df-0g 17460  df-gsum 17461  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-submnd 18767  df-frmd 18832  df-mrex 35513  df-mex 35514  df-mdv 35515  df-mrsub 35517  df-msub 35518  df-mvh 35519  df-mpst 35520  df-msr 35521  df-msta 35522  df-mfs 35523  df-mcls 35524  df-mpps 35525  df-mthm 35526

This theorem is referenced by: (None)