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Theorem msubco 32298
 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 2772 . . . . 5 (mEx‘𝑇) = (mEx‘𝑇)
2 eqid 2772 . . . . 5 (mRSubst‘𝑇) = (mRSubst‘𝑇)
3 msubco.s . . . . 5 𝑆 = (mSubst‘𝑇)
41, 2, 3elmsubrn 32295 . . . 4 ran 𝑆 = ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩))
54eleq2i 2851 . . 3 (𝐹 ∈ ran 𝑆𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩)))
6 eqid 2772 . . . 4 (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩)) = (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩))
7 fvex 6506 . . . . 5 (mEx‘𝑇) ∈ V
87mptex 6806 . . . 4 (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∈ V
96, 8elrnmpti 5669 . . 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 267 . 2 (𝐹 ∈ ran 𝑆 ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩))
111, 2, 3elmsubrn 32295 . . . 4 ran 𝑆 = ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩))
1211eleq2i 2851 . . 3 (𝐺 ∈ ran 𝑆𝐺 ∈ ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)))
13 eqid 2772 . . . 4 (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) = (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩))
147mptex 6806 . . . 4 (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩) ∈ V
1513, 14elrnmpti 5669 . . 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 267 . 2 (𝐺 ∈ ran 𝑆 ↔ ∃𝑔 ∈ ran (mRSubst‘𝑇)𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩))
17 reeanv 3302 . . 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 477 . . . . . . . . . . . 12 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑦 ∈ (mEx‘𝑇))
19 eqid 2772 . . . . . . . . . . . . 13 (mTC‘𝑇) = (mTC‘𝑇)
20 eqid 2772 . . . . . . . . . . . . 13 (mREx‘𝑇) = (mREx‘𝑇)
2119, 1, 20mexval 32269 . . . . . . . . . . . 12 (mEx‘𝑇) = ((mTC‘𝑇) × (mREx‘𝑇))
2218, 21syl6eleq 2870 . . . . . . . . . . 11 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑦 ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
23 xp1st 7527 . . . . . . . . . . 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 32284 . . . . . . . . . . . 12 (𝑔 ∈ ran (mRSubst‘𝑇) → 𝑔:(mREx‘𝑇)⟶(mREx‘𝑇))
2625ad2antlr 714 . . . . . . . . . . 11 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑔:(mREx‘𝑇)⟶(mREx‘𝑇))
27 xp2nd 7528 . . . . . . . . . . . 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 6671 . . . . . . . . . 10 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → (𝑔‘(2nd𝑦)) ∈ (mREx‘𝑇))
30 opelxpi 5438 . . . . . . . . . 10 (((1st𝑦) ∈ (mTC‘𝑇) ∧ (𝑔‘(2nd𝑦)) ∈ (mREx‘𝑇)) → ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
3124, 29, 30syl2anc 576 . . . . . . . . 9 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
3231, 21syl6eleqr 2871 . . . . . . . 8 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ ∈ (mEx‘𝑇))
33 eqidd 2773 . . . . . . . 8 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩))
34 eqidd 2773 . . . . . . . 8 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩))
35 fvex 6506 . . . . . . . . . 10 (1st𝑦) ∈ V
36 fvex 6506 . . . . . . . . . 10 (𝑔‘(2nd𝑦)) ∈ V
3735, 36op1std 7505 . . . . . . . . 9 (𝑥 = ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ → (1st𝑥) = (1st𝑦))
3835, 36op2ndd 7506 . . . . . . . . . 10 (𝑥 = ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ → (2nd𝑥) = (𝑔‘(2nd𝑦)))
3938fveq2d 6497 . . . . . . . . 9 (𝑥 = ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ → (𝑓‘(2nd𝑥)) = (𝑓‘(𝑔‘(2nd𝑦))))
4037, 39opeq12d 4679 . . . . . . . 8 (𝑥 = ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ → ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩ = ⟨(1st𝑦), (𝑓‘(𝑔‘(2nd𝑦)))⟩)
4132, 33, 34, 40fmptco 6708 . . . . . . 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 6582 . . . . . . . . . 10 ((𝑔:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ (2nd𝑦) ∈ (mREx‘𝑇)) → ((𝑓𝑔)‘(2nd𝑦)) = (𝑓‘(𝑔‘(2nd𝑦))))
4326, 28, 42syl2anc 576 . . . . . . . . 9 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ((𝑓𝑔)‘(2nd𝑦)) = (𝑓‘(𝑔‘(2nd𝑦))))
4443opeq2d 4678 . . . . . . . 8 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩ = ⟨(1st𝑦), (𝑓‘(𝑔‘(2nd𝑦)))⟩)
4544mpteq2dva 5016 . . . . . . 7 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑓‘(𝑔‘(2nd𝑦)))⟩))
4641, 45eqtr4d 2811 . . . . . 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 32288 . . . . . . . 8 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑓𝑔) ∈ ran (mRSubst‘𝑇))
487mptex 6806 . . . . . . . 8 (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩) ∈ V
49 eqid 2772 . . . . . . . . 9 ( ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (‘(2nd𝑦))⟩)) = ( ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (‘(2nd𝑦))⟩))
50 fveq1 6492 . . . . . . . . . . 11 ( = (𝑓𝑔) → (‘(2nd𝑦)) = ((𝑓𝑔)‘(2nd𝑦)))
5150opeq2d 4678 . . . . . . . . . 10 ( = (𝑓𝑔) → ⟨(1st𝑦), (‘(2nd𝑦))⟩ = ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩)
5251mpteq2dv 5017 . . . . . . . . 9 ( = (𝑓𝑔) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (‘(2nd𝑦))⟩) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩))
5349, 52elrnmpt1s 5666 . . . . . . . 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 577 . . . . . . 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 32295 . . . . . . 7 ran 𝑆 = ran ( ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (‘(2nd𝑦))⟩))
5654, 55syl6eleqr 2871 . . . . . 6 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩) ∈ ran 𝑆)
5746, 56eqeltrd 2860 . . . . 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 5572 . . . . . . 7 (𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) → (𝐹𝐺) = ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∘ 𝐺))
59 coeq2 5573 . . . . . . 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 2828 . . . . . 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 2844 . . . . 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 239 . . . 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 3231 . . 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 227 . 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 588 1 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → (𝐹𝐺) ∈ ran 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2050  ∃wrex 3083  Vcvv 3409  ⟨cop 4441   ↦ cmpt 5002   × cxp 5399  ran crn 5402   ∘ ccom 5405  ⟶wf 6178  ‘cfv 6182  1st c1st 7493  2nd c2nd 7494  mTCcmtc 32231  mRExcmrex 32233  mExcmex 32234  mRSubstcmrsub 32237  mSubstcmsub 32238 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10385  ax-resscn 10386  ax-1cn 10387  ax-icn 10388  ax-addcl 10389  ax-addrcl 10390  ax-mulcl 10391  ax-mulrcl 10392  ax-mulcom 10393  ax-addass 10394  ax-mulass 10395  ax-distr 10396  ax-i2m1 10397  ax-1ne0 10398  ax-1rid 10399  ax-rnegex 10400  ax-rrecex 10401  ax-cnre 10402  ax-pre-lttri 10403  ax-pre-lttrn 10404  ax-pre-ltadd 10405  ax-pre-mulgt0 10406 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5306  df-eprel 5311  df-po 5320  df-so 5321  df-fr 5360  df-we 5362  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7495  df-2nd 7496  df-wrecs 7744  df-recs 7806  df-rdg 7844  df-1o 7899  df-oadd 7903  df-er 8083  df-map 8202  df-pm 8203  df-en 8301  df-dom 8302  df-sdom 8303  df-fin 8304  df-card 9156  df-pnf 10470  df-mnf 10471  df-xr 10472  df-ltxr 10473  df-le 10474  df-sub 10666  df-neg 10667  df-nn 11434  df-2 11497  df-n0 11702  df-xnn0 11774  df-z 11788  df-uz 12053  df-fz 12703  df-fzo 12844  df-seq 13179  df-hash 13500  df-word 13667  df-lsw 13720  df-concat 13728  df-s1 13753  df-substr 13798  df-pfx 13847  df-struct 16335  df-ndx 16336  df-slot 16337  df-base 16339  df-sets 16340  df-ress 16341  df-plusg 16428  df-0g 16565  df-gsum 16566  df-mgm 17704  df-sgrp 17746  df-mnd 17757  df-mhm 17797  df-submnd 17798  df-frmd 17849  df-vrmd 17850  df-mrex 32253  df-mex 32254  df-mrsub 32257  df-msub 32258 This theorem is referenced by:  mclsppslem  32350
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