 Mathbox for Mario Carneiro < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  msubco Structured version   Visualization version   GIF version

Theorem msubco 31766
 Description: The composition of two substitutions is a substitution. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypothesis
Ref Expression
msubco.s 𝑆 = (mSubst‘𝑇)
Assertion
Ref Expression
msubco ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → (𝐹𝐺) ∈ ran 𝑆)

Proof of Theorem msubco
Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2771 . . . . 5 (mEx‘𝑇) = (mEx‘𝑇)
2 eqid 2771 . . . . 5 (mRSubst‘𝑇) = (mRSubst‘𝑇)
3 msubco.s . . . . 5 𝑆 = (mSubst‘𝑇)
41, 2, 3elmsubrn 31763 . . . 4 ran 𝑆 = ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩))
54eleq2i 2842 . . 3 (𝐹 ∈ ran 𝑆𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩)))
6 eqid 2771 . . . 4 (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩)) = (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩))
7 fvex 6344 . . . . 5 (mEx‘𝑇) ∈ V
87mptex 6633 . . . 4 (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∈ V
96, 8elrnmpti 5513 . . 3 (𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩)) ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩))
105, 9bitri 264 . 2 (𝐹 ∈ ran 𝑆 ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩))
111, 2, 3elmsubrn 31763 . . . 4 ran 𝑆 = ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩))
1211eleq2i 2842 . . 3 (𝐺 ∈ ran 𝑆𝐺 ∈ ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)))
13 eqid 2771 . . . 4 (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) = (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩))
147mptex 6633 . . . 4 (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩) ∈ V
1513, 14elrnmpti 5513 . . 3 (𝐺 ∈ ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) ↔ ∃𝑔 ∈ ran (mRSubst‘𝑇)𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩))
1612, 15bitri 264 . 2 (𝐺 ∈ ran 𝑆 ↔ ∃𝑔 ∈ ran (mRSubst‘𝑇)𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩))
17 reeanv 3255 . . 3 (∃𝑓 ∈ ran (mRSubst‘𝑇)∃𝑔 ∈ ran (mRSubst‘𝑇)(𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∧ 𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) ↔ (∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∧ ∃𝑔 ∈ ran (mRSubst‘𝑇)𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)))
18 simpr 471 . . . . . . . . . . . 12 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑦 ∈ (mEx‘𝑇))
19 eqid 2771 . . . . . . . . . . . . 13 (mTC‘𝑇) = (mTC‘𝑇)
20 eqid 2771 . . . . . . . . . . . . 13 (mREx‘𝑇) = (mREx‘𝑇)
2119, 1, 20mexval 31737 . . . . . . . . . . . 12 (mEx‘𝑇) = ((mTC‘𝑇) × (mREx‘𝑇))
2218, 21syl6eleq 2860 . . . . . . . . . . 11 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑦 ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
23 xp1st 7351 . . . . . . . . . . 11 (𝑦 ∈ ((mTC‘𝑇) × (mREx‘𝑇)) → (1st𝑦) ∈ (mTC‘𝑇))
2422, 23syl 17 . . . . . . . . . 10 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → (1st𝑦) ∈ (mTC‘𝑇))
252, 20mrsubf 31752 . . . . . . . . . . . 12 (𝑔 ∈ ran (mRSubst‘𝑇) → 𝑔:(mREx‘𝑇)⟶(mREx‘𝑇))
2625ad2antlr 706 . . . . . . . . . . 11 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑔:(mREx‘𝑇)⟶(mREx‘𝑇))
27 xp2nd 7352 . . . . . . . . . . . 12 (𝑦 ∈ ((mTC‘𝑇) × (mREx‘𝑇)) → (2nd𝑦) ∈ (mREx‘𝑇))
2822, 27syl 17 . . . . . . . . . . 11 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → (2nd𝑦) ∈ (mREx‘𝑇))
2926, 28ffvelrnd 6505 . . . . . . . . . 10 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → (𝑔‘(2nd𝑦)) ∈ (mREx‘𝑇))
30 opelxpi 5287 . . . . . . . . . 10 (((1st𝑦) ∈ (mTC‘𝑇) ∧ (𝑔‘(2nd𝑦)) ∈ (mREx‘𝑇)) → ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
3124, 29, 30syl2anc 573 . . . . . . . . 9 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
3231, 21syl6eleqr 2861 . . . . . . . 8 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ ∈ (mEx‘𝑇))
33 eqidd 2772 . . . . . . . 8 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩))
34 eqidd 2772 . . . . . . . 8 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩))
35 fvex 6344 . . . . . . . . . 10 (1st𝑦) ∈ V
36 fvex 6344 . . . . . . . . . 10 (𝑔‘(2nd𝑦)) ∈ V
3735, 36op1std 7329 . . . . . . . . 9 (𝑥 = ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ → (1st𝑥) = (1st𝑦))
3835, 36op2ndd 7330 . . . . . . . . . 10 (𝑥 = ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ → (2nd𝑥) = (𝑔‘(2nd𝑦)))
3938fveq2d 6337 . . . . . . . . 9 (𝑥 = ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ → (𝑓‘(2nd𝑥)) = (𝑓‘(𝑔‘(2nd𝑦))))
4037, 39opeq12d 4548 . . . . . . . 8 (𝑥 = ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ → ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩ = ⟨(1st𝑦), (𝑓‘(𝑔‘(2nd𝑦)))⟩)
4132, 33, 34, 40fmptco 6542 . . . . . . 7 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∘ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑓‘(𝑔‘(2nd𝑦)))⟩))
42 fvco3 6419 . . . . . . . . . 10 ((𝑔:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ (2nd𝑦) ∈ (mREx‘𝑇)) → ((𝑓𝑔)‘(2nd𝑦)) = (𝑓‘(𝑔‘(2nd𝑦))))
4326, 28, 42syl2anc 573 . . . . . . . . 9 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ((𝑓𝑔)‘(2nd𝑦)) = (𝑓‘(𝑔‘(2nd𝑦))))
4443opeq2d 4547 . . . . . . . 8 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩ = ⟨(1st𝑦), (𝑓‘(𝑔‘(2nd𝑦)))⟩)
4544mpteq2dva 4879 . . . . . . 7 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑓‘(𝑔‘(2nd𝑦)))⟩))
4641, 45eqtr4d 2808 . . . . . 6 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∘ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩))
472mrsubco 31756 . . . . . . . 8 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑓𝑔) ∈ ran (mRSubst‘𝑇))
487mptex 6633 . . . . . . . 8 (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩) ∈ V
49 eqid 2771 . . . . . . . . 9 ( ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (‘(2nd𝑦))⟩)) = ( ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (‘(2nd𝑦))⟩))
50 fveq1 6332 . . . . . . . . . . 11 ( = (𝑓𝑔) → (‘(2nd𝑦)) = ((𝑓𝑔)‘(2nd𝑦)))
5150opeq2d 4547 . . . . . . . . . 10 ( = (𝑓𝑔) → ⟨(1st𝑦), (‘(2nd𝑦))⟩ = ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩)
5251mpteq2dv 4880 . . . . . . . . 9 ( = (𝑓𝑔) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (‘(2nd𝑦))⟩) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩))
5349, 52elrnmpt1s 5510 . . . . . . . 8 (((𝑓𝑔) ∈ ran (mRSubst‘𝑇) ∧ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩) ∈ V) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩) ∈ ran ( ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (‘(2nd𝑦))⟩)))
5447, 48, 53sylancl 574 . . . . . . 7 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩) ∈ ran ( ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (‘(2nd𝑦))⟩)))
551, 2, 3elmsubrn 31763 . . . . . . 7 ran 𝑆 = ran ( ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (‘(2nd𝑦))⟩))
5654, 55syl6eleqr 2861 . . . . . 6 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩) ∈ ran 𝑆)
5746, 56eqeltrd 2850 . . . . 5 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∘ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) ∈ ran 𝑆)
58 coeq1 5417 . . . . . . 7 (𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) → (𝐹𝐺) = ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∘ 𝐺))
59 coeq2 5418 . . . . . . 7 (𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩) → ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∘ 𝐺) = ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∘ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)))
6058, 59sylan9eq 2825 . . . . . 6 ((𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∧ 𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) → (𝐹𝐺) = ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∘ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)))
6160eleq1d 2835 . . . . 5 ((𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∧ 𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) → ((𝐹𝐺) ∈ ran 𝑆 ↔ ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∘ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) ∈ ran 𝑆))
6257, 61syl5ibrcom 237 . . . 4 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → ((𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∧ 𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) → (𝐹𝐺) ∈ ran 𝑆))
6362rexlimivv 3184 . . 3 (∃𝑓 ∈ ran (mRSubst‘𝑇)∃𝑔 ∈ ran (mRSubst‘𝑇)(𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∧ 𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) → (𝐹𝐺) ∈ ran 𝑆)
6417, 63sylbir 225 . 2 ((∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∧ ∃𝑔 ∈ ran (mRSubst‘𝑇)𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) → (𝐹𝐺) ∈ ran 𝑆)
6510, 16, 64syl2anb 585 1 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → (𝐹𝐺) ∈ ran 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∃wrex 3062  Vcvv 3351  ⟨cop 4323   ↦ cmpt 4864   × cxp 5248  ran crn 5251   ∘ ccom 5254  ⟶wf 6026  ‘cfv 6030  1st c1st 7317  2nd c2nd 7318  mTCcmtc 31699  mRExcmrex 31701  mExcmex 31702  mRSubstcmrsub 31705  mSubstcmsub 31706 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7900  df-map 8015  df-pm 8016  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-card 8969  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-2 11285  df-n0 11500  df-xnn0 11571  df-z 11585  df-uz 11894  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-word 13495  df-lsw 13496  df-concat 13497  df-s1 13498  df-substr 13499  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-0g 16310  df-gsum 16311  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-mhm 17543  df-submnd 17544  df-frmd 17594  df-vrmd 17595  df-mrex 31721  df-mex 31722  df-mrsub 31725  df-msub 31726 This theorem is referenced by:  mclsppslem  31818
 Copyright terms: Public domain W3C validator