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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
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