mthmpps - Mathbox for Mario Carneiro

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Theorem mthmpps 36347

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 2760	. . . . . . . . . . . . . 14 ⊢ (mPreSt‘𝑇) = (mPreSt‘𝑇)
4	2, 3	mthmsta 36343	. . . . . . . . . . . . 13 ⊢ 𝑈 ⊆ (mPreSt‘𝑇)
5		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈)
6	4, 5	sselid 3906	. . . . . . . . . . . 12 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝐶, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇))
7		mthmpps.d	. . . . . . . . . . . . 13 ⊢ 𝐷 = (mDV‘𝑇)
8		eqid 2760	. . . . . . . . . . . . 13 ⊢ (mEx‘𝑇) = (mEx‘𝑇)
9	7, 8, 3	elmpst 36301	. . . . . . . . . . . 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 1160	. . . . . . . . . 10 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶 ⊆ 𝐷 ∧ ^◡𝐶 = 𝐶))
12	11	simpld 500	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐶 ⊆ 𝐷)
13		difssd 4073	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐷 ∖ (𝑍 × 𝑍)) ⊆ 𝐷)
14	12, 13	unssd 4190	. . . . . . . 8 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ⊆ 𝐷)
15	1, 14	eqsstrid 3946	. . . . . . 7 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝑀 ⊆ 𝐷)
16	11	simprd 501	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ^◡𝐶 = 𝐶)
17		cnvdif 6133	. . . . . . . . . . 11 ⊢ ^◡(𝐷 ∖ (𝑍 × 𝑍)) = (^◡𝐷 ∖ ^◡(𝑍 × 𝑍))
18		cnvdif 6133	. . . . . . . . . . . . . 14 ⊢ ^◡(((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (^◡((mVR‘𝑇) × (mVR‘𝑇)) ∖ ^◡ I )
19		cnvxp 6146	. . . . . . . . . . . . . . 15 ⊢ ^◡((mVR‘𝑇) × (mVR‘𝑇)) = ((mVR‘𝑇) × (mVR‘𝑇))
20		cnvi 5862	. . . . . . . . . . . . . . 15 ⊢ ^◡ I = I
21	19, 20	difeq12i 4061	. . . . . . . . . . . . . 14 ⊢ (^◡((mVR‘𝑇) × (mVR‘𝑇)) ∖ ^◡ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
22	18, 21	eqtri 2783	. . . . . . . . . . . . 13 ⊢ ^◡(((mVR‘𝑇) × (mVR‘𝑇)) ∖ I ) = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
23		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
24	23, 7	mdvval 36269	. . . . . . . . . . . . . 14 ⊢ 𝐷 = (((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
25	24	cnveqi 5851	. . . . . . . . . . . . 13 ⊢ ^◡𝐷 = ^◡(((mVR‘𝑇) × (mVR‘𝑇)) ∖ I )
26	22, 25, 24	3eqtr4i 2793	. . . . . . . . . . . 12 ⊢ ^◡𝐷 = 𝐷
27		cnvxp 6146	. . . . . . . . . . . 12 ⊢ ^◡(𝑍 × 𝑍) = (𝑍 × 𝑍)
28	26, 27	difeq12i 4061	. . . . . . . . . . 11 ⊢ (^◡𝐷 ∖ ^◡(𝑍 × 𝑍)) = (𝐷 ∖ (𝑍 × 𝑍))
29	17, 28	eqtri 2783	. . . . . . . . . 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 4168	. . . . . . . 8 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (^◡𝐶 ∪ ^◡(𝐷 ∖ (𝑍 × 𝑍))) = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))))
32	1	cnveqi 5851	. . . . . . . . 9 ⊢ ^◡𝑀 = ^◡(𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍)))
33		cnvun 6132	. . . . . . . . 9 ⊢ ^◡(𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) = (^◡𝐶 ∪ ^◡(𝐷 ∖ (𝑍 × 𝑍)))
34	32, 33	eqtri 2783	. . . . . . . 8 ⊢ ^◡𝑀 = (^◡𝐶 ∪ ^◡(𝐷 ∖ (𝑍 × 𝑍)))
35	31, 34, 1	3eqtr4g 2820	. . . . . . 7 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ^◡𝑀 = 𝑀)
36	15, 35	jca 521	. . . . . 6 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑀 ⊆ 𝐷 ∧ ^◡𝑀 = 𝑀))
37	10	simp2d 1161	. . . . . 6 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin))
38	10	simp3d 1162	. . . . . 6 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐴 ∈ (mEx‘𝑇))
39	7, 8, 3	elmpst 36301	. . . . . 6 ⊢ (⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ ((𝑀 ⊆ 𝐷 ∧ ^◡𝑀 = 𝑀) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
40	36, 37, 38, 39	syl3anbrc 1362	. . . . 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 36341	. . . . . . 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 2760	. . . . . . . 8 ⊢ (mCls‘𝑇) = (mCls‘𝑇)
46		simpll 779	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑇 ∈ mFS)
47	15	adantr 486	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑀 ⊆ 𝐷)
48	37	simpld 500	. . . . . . . . 9 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐻 ⊆ (mEx‘𝑇))
49	48	adantr 486	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐻 ⊆ (mEx‘𝑇))
50	3, 42	mppspst 36339	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐽 ⊆ (mPreSt‘𝑇)
51		simprl 783	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 ∈ 𝐽)
52	50, 51	sselid 3906	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 ∈ (mPreSt‘𝑇))
53	3	mpst123 36305	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (mPreSt‘𝑇) → 𝑥 = ⟨(1^st ‘(1^st ‘𝑥)), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩)
54	52, 53	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 = ⟨(1^st ‘(1^st ‘𝑥)), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩)
55	54	fveq2d 6889	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅‘𝑥) = (𝑅‘⟨(1^st ‘(1^st ‘𝑥)), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩))
56		simprr 785	. . . . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅‘⟨(1^st ‘(1^st ‘𝑥)), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
58	54, 52	eqeltrrd 2861	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1^st ‘(1^st ‘𝑥)), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩ ∈ (mPreSt‘𝑇))
59		mthmpps.v	. . . . . . . . . . . . . . . . 17 ⊢ 𝑉 = (mVars‘𝑇)
60		eqid 2760	. . . . . . . . . . . . . . . . 17 ⊢ ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) = ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)}))
61	59, 3, 41, 60	msrval 36303	. . . . . . . . . . . . . . . 16 ⊢ (⟨(1^st ‘(1^st ‘𝑥)), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩ ∈ (mPreSt‘𝑇) → (𝑅‘⟨(1^st ‘(1^st ‘𝑥)), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩) = ⟨((1^st ‘(1^st ‘𝑥)) ∩ (∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) × ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})))), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩)
62	58, 61	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑅‘⟨(1^st ‘(1^st ‘𝑥)), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩) = ⟨((1^st ‘(1^st ‘𝑥)) ∩ (∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) × ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})))), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩)
63		mthmpps.z	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑍 = ∪ (𝑉 “ (𝐻 ∪ {𝐴}))
64	59, 3, 41, 63	msrval 36303	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝐶, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → (𝑅‘⟨𝐶, 𝐻, 𝐴⟩) = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
65	6, 64	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑅‘⟨𝐶, 𝐻, 𝐴⟩) = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
66	65	adantr 486	. . . . . . . . . . . . . . 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 2803	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨((1^st ‘(1^st ‘𝑥)) ∩ (∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) × ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})))), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩ = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
68		fvex 6898	. . . . . . . . . . . . . . . 16 ⊢ (1^st ‘(1^st ‘𝑥)) ∈ V
69	68	inex1 5276	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘(1^st ‘𝑥)) ∩ (∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) × ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})))) ∈ V
70		fvex 6898	. . . . . . . . . . . . . . 15 ⊢ (2^nd ‘(1^st ‘𝑥)) ∈ V
71		fvex 6898	. . . . . . . . . . . . . . 15 ⊢ (2^nd ‘𝑥) ∈ V
72	69, 70, 71	otth 5452	. . . . . . . . . . . . . 14 ⊢ (⟨((1^st ‘(1^st ‘𝑥)) ∩ (∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) × ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})))), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩ = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ ↔ (((1^st ‘(1^st ‘𝑥)) ∩ (∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) × ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})))) = (𝐶 ∩ (𝑍 × 𝑍)) ∧ (2^nd ‘(1^st ‘𝑥)) = 𝐻 ∧ (2^nd ‘𝑥) = 𝐴))
73	67, 72	sylib 221	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (((1^st ‘(1^st ‘𝑥)) ∩ (∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) × ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})))) = (𝐶 ∩ (𝑍 × 𝑍)) ∧ (2^nd ‘(1^st ‘𝑥)) = 𝐻 ∧ (2^nd ‘𝑥) = 𝐴))
74	73	simp1d 1160	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1^st ‘(1^st ‘𝑥)) ∩ (∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) × ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})))) = (𝐶 ∩ (𝑍 × 𝑍)))
75	73	simp2d 1161	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (2^nd ‘(1^st ‘𝑥)) = 𝐻)
76	73	simp3d 1162	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (2^nd ‘𝑥) = 𝐴)
77	76	sneqd 4595	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → {(2^nd ‘𝑥)} = {𝐴})
78	75, 77	uneq12d 4168	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)}) = (𝐻 ∪ {𝐴}))
79	78	imaeq2d 6051	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) = (𝑉 “ (𝐻 ∪ {𝐴})))
80	79	unieqd 4879	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) = ∪ (𝑉 “ (𝐻 ∪ {𝐴})))
81	80, 63	eqtr4di 2813	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) = 𝑍)
82	81	sqxpeqd 5682	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) × ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)}))) = (𝑍 × 𝑍))
83	82	ineq2d 4117	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1^st ‘(1^st ‘𝑥)) ∩ (∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})) × ∪ (𝑉 “ ((2^nd ‘(1^st ‘𝑥)) ∪ {(2^nd ‘𝑥)})))) = ((1^st ‘(1^st ‘𝑥)) ∩ (𝑍 × 𝑍)))
84	74, 83	eqtr3d 2797	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (𝐶 ∩ (𝑍 × 𝑍)) = ((1^st ‘(1^st ‘𝑥)) ∩ (𝑍 × 𝑍)))
85		inss1 4133	. . . . . . . . . . 11 ⊢ (𝐶 ∩ (𝑍 × 𝑍)) ⊆ 𝐶
86	84, 85	eqsstrrdi 3953	. . . . . . . . . 10 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1^st ‘(1^st ‘𝑥)) ∩ (𝑍 × 𝑍)) ⊆ 𝐶)
87		eqidd 2761	. . . . . . . . . . . . . . 15 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1^st ‘(1^st ‘𝑥)) = (1^st ‘(1^st ‘𝑥)))
88	87, 75, 76	oteq123d 4847	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1^st ‘(1^st ‘𝑥)), (2^nd ‘(1^st ‘𝑥)), (2^nd ‘𝑥)⟩ = ⟨(1^st ‘(1^st ‘𝑥)), 𝐻, 𝐴⟩)
89	54, 88	eqtrd 2795	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝑥 = ⟨(1^st ‘(1^st ‘𝑥)), 𝐻, 𝐴⟩)
90	89, 52	eqeltrrd 2861	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1^st ‘(1^st ‘𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇))
91	7, 8, 3	elmpst 36301	. . . . . . . . . . . . . 14 ⊢ (⟨(1^st ‘(1^st ‘𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ↔ (((1^st ‘(1^st ‘𝑥)) ⊆ 𝐷 ∧ ^◡(1^st ‘(1^st ‘𝑥)) = (1^st ‘(1^st ‘𝑥))) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
92	91	simp1bi 1163	. . . . . . . . . . . . 13 ⊢ (⟨(1^st ‘(1^st ‘𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → ((1^st ‘(1^st ‘𝑥)) ⊆ 𝐷 ∧ ^◡(1^st ‘(1^st ‘𝑥)) = (1^st ‘(1^st ‘𝑥))))
93	92	simpld 500	. . . . . . . . . . . 12 ⊢ (⟨(1^st ‘(1^st ‘𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → (1^st ‘(1^st ‘𝑥)) ⊆ 𝐷)
94	90, 93	syl 18	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1^st ‘(1^st ‘𝑥)) ⊆ 𝐷)
95	94	ssdifd 4081	. . . . . . . . . 10 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1^st ‘(1^st ‘𝑥)) ∖ (𝑍 × 𝑍)) ⊆ (𝐷 ∖ (𝑍 × 𝑍)))
96		unss12 4186	. . . . . . . . . 10 ⊢ ((((1^st ‘(1^st ‘𝑥)) ∩ (𝑍 × 𝑍)) ⊆ 𝐶 ∧ ((1^st ‘(1^st ‘𝑥)) ∖ (𝑍 × 𝑍)) ⊆ (𝐷 ∖ (𝑍 × 𝑍))) → (((1^st ‘(1^st ‘𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1^st ‘(1^st ‘𝑥)) ∖ (𝑍 × 𝑍))) ⊆ (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))))
97	86, 95, 96	syl2anc 596	. . . . . . . . 9 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (((1^st ‘(1^st ‘𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1^st ‘(1^st ‘𝑥)) ∖ (𝑍 × 𝑍))) ⊆ (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))))
98		inundif 4434	. . . . . . . . . 10 ⊢ (((1^st ‘(1^st ‘𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1^st ‘(1^st ‘𝑥)) ∖ (𝑍 × 𝑍))) = (1^st ‘(1^st ‘𝑥))
99	98	eqcomi 2769	. . . . . . . . 9 ⊢ (1^st ‘(1^st ‘𝑥)) = (((1^st ‘(1^st ‘𝑥)) ∩ (𝑍 × 𝑍)) ∪ ((1^st ‘(1^st ‘𝑥)) ∖ (𝑍 × 𝑍)))
100	97, 99, 1	3sstr4g 3961	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → (1^st ‘(1^st ‘𝑥)) ⊆ 𝑀)
101		ssidd 3931	. . . . . . . 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 36333	. . . . . . 7 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ((1^st ‘(1^st ‘𝑥))(mCls‘𝑇)𝐻) ⊆ (𝑀(mCls‘𝑇)𝐻))
103	89, 51	eqeltrrd 2861	. . . . . . . 8 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → ⟨(1^st ‘(1^st ‘𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽)
104	3, 42, 45	elmpps 36338	. . . . . . . . 9 ⊢ (⟨(1^st ‘(1^st ‘𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨(1^st ‘(1^st ‘𝑥)), 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ∧ 𝐴 ∈ ((1^st ‘(1^st ‘𝑥))(mCls‘𝑇)𝐻)))
105	104	simprbi 503	. . . . . . . 8 ⊢ (⟨(1^st ‘(1^st ‘𝑥)), 𝐻, 𝐴⟩ ∈ 𝐽 → 𝐴 ∈ ((1^st ‘(1^st ‘𝑥))(mCls‘𝑇)𝐻))
106	103, 105	syl 18	. . . . . . 7 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐴 ∈ ((1^st ‘(1^st ‘𝑥))(mCls‘𝑇)𝐻))
107	102, 106	sseldd 3909	. . . . . 6 ⊢ (((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) ∧ (𝑥 ∈ 𝐽 ∧ (𝑅‘𝑥) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))) → 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻))
108	44, 107	rexlimddv 3169	. . . . 5 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻))
109	3, 42, 45	elmpps 36338	. . . . 5 ⊢ (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) ∧ 𝐴 ∈ (𝑀(mCls‘𝑇)𝐻)))
110	40, 108, 109	sylanbrc 595	. . . 4 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽)
111	1	ineq1i 4113	. . . . . . . 8 ⊢ (𝑀 ∩ (𝑍 × 𝑍)) = ((𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ∩ (𝑍 × 𝑍))
112		indir 4231	. . . . . . . 8 ⊢ ((𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ∩ (𝑍 × 𝑍)) = ((𝐶 ∩ (𝑍 × 𝑍)) ∪ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)))
113		disjdifr 4426	. . . . . . . . . 10 ⊢ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) = ∅
114		0ss 4349	. . . . . . . . . 10 ⊢ ∅ ⊆ (𝐶 ∩ (𝑍 × 𝑍))
115	113, 114	eqsstri 3954	. . . . . . . . 9 ⊢ ((𝐷 ∖ (𝑍 × 𝑍)) ∩ (𝑍 × 𝑍)) ⊆ (𝐶 ∩ (𝑍 × 𝑍))
116		ssequn2 4187	. . . . . . . . 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 2787	. . . . . . 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 4844	. . . . 5 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → ⟨(𝑀 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ = ⟨(𝐶 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
121	59, 3, 41, 63	msrval 36303	. . . . . 6 ⊢ (⟨𝑀, 𝐻, 𝐴⟩ ∈ (mPreSt‘𝑇) → (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = ⟨(𝑀 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
122	40, 121	syl 18	. . . . 5 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = ⟨(𝑀 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
123	120, 122, 65	3eqtr4d 2805	. . . 4 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))
124	110, 123	jca 521	. . 3 ⊢ ((𝑇 ∈ mFS ∧ ⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈) → (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩)))
125	124	ex 418	. 2 ⊢ (𝑇 ∈ mFS → (⟨𝐶, 𝐻, 𝐴⟩ ∈ 𝑈 → (⟨𝑀, 𝐻, 𝐴⟩ ∈ 𝐽 ∧ (𝑅‘⟨𝑀, 𝐻, 𝐴⟩) = (𝑅‘⟨𝐶, 𝐻, 𝐴⟩))))
126	41, 42, 2	mthmi 36342	. 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⟩))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∃wrex 3086 ⊆ wss 3895 ∖ cdif 4045 ∩ cin 4089 ∪ cun 4149 ∅c0 4278 {csn 4583 ⟨cotp 4591 ∪ cuni 4866 I cid 5544 × cxp 5648 ^◡ccnv 5649 “ cima 5653 ‘cfv 6538 (class class class)co 7420 1^st c1st 7999 2^nd c2nd 8000 Fincfn 8973 mVRcmvar 36226 mExcmex 36232 mDVcmdv 36233 mVarscmvrs 36234 mPreStcmpst 36238 mStRedcmsr 36239 mFScmfs 36241 mClscmcls 36242 mPPStcmpps 36243 mThmcmthm 36244

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-ot 4592 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-er 8717 df-map 8849 df-pm 8850 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-2 12405 df-n0 12607 df-z 12694 df-uz 12966 df-fz 13640 df-fzo 13789 df-seq 14145 df-hash 14475 df-word 14659 df-concat 14716 df-s1 14743 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-0g 17612 df-gsum 17613 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-frmd 19045 df-mrex 36251 df-mex 36252 df-mdv 36253 df-mrsub 36255 df-msub 36256 df-mvh 36257 df-mpst 36258 df-msr 36259 df-msta 36260 df-mfs 36261 df-mcls 36262 df-mpps 36263 df-mthm 36264

This theorem is used by: (None)

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