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Theorem msubco 35515
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 2734 . . . . 5 (mEx‘𝑇) = (mEx‘𝑇)
2 eqid 2734 . . . . 5 (mRSubst‘𝑇) = (mRSubst‘𝑇)
3 msubco.s . . . . 5 𝑆 = (mSubst‘𝑇)
41, 2, 3elmsubrn 35512 . . . 4 ran 𝑆 = ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩))
54eleq2i 2830 . . 3 (𝐹 ∈ ran 𝑆𝐹 ∈ ran (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩)))
6 eqid 2734 . . . 4 (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩)) = (𝑓 ∈ ran (mRSubst‘𝑇) ↦ (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩))
7 fvex 6919 . . . . 5 (mEx‘𝑇) ∈ V
87mptex 7242 . . . 4 (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∈ V
96, 8elrnmpti 5975 . . 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 275 . 2 (𝐹 ∈ ran 𝑆 ↔ ∃𝑓 ∈ ran (mRSubst‘𝑇)𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩))
111, 2, 3elmsubrn 35512 . . . 4 ran 𝑆 = ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩))
1211eleq2i 2830 . . 3 (𝐺 ∈ ran 𝑆𝐺 ∈ ran (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)))
13 eqid 2734 . . . 4 (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩)) = (𝑔 ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩))
147mptex 7242 . . . 4 (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩) ∈ V
1513, 14elrnmpti 5975 . . 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 275 . 2 (𝐺 ∈ ran 𝑆 ↔ ∃𝑔 ∈ ran (mRSubst‘𝑇)𝐺 = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩))
17 reeanv 3226 . . 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 484 . . . . . . . . . . . 12 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑦 ∈ (mEx‘𝑇))
19 eqid 2734 . . . . . . . . . . . . 13 (mTC‘𝑇) = (mTC‘𝑇)
20 eqid 2734 . . . . . . . . . . . . 13 (mREx‘𝑇) = (mREx‘𝑇)
2119, 1, 20mexval 35486 . . . . . . . . . . . 12 (mEx‘𝑇) = ((mTC‘𝑇) × (mREx‘𝑇))
2218, 21eleqtrdi 2848 . . . . . . . . . . 11 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑦 ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
23 xp1st 8044 . . . . . . . . . . 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 35501 . . . . . . . . . . . 12 (𝑔 ∈ ran (mRSubst‘𝑇) → 𝑔:(mREx‘𝑇)⟶(mREx‘𝑇))
2625ad2antlr 727 . . . . . . . . . . 11 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → 𝑔:(mREx‘𝑇)⟶(mREx‘𝑇))
27 xp2nd 8045 . . . . . . . . . . . 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, 28ffvelcdmd 7104 . . . . . . . . . 10 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → (𝑔‘(2nd𝑦)) ∈ (mREx‘𝑇))
30 opelxpi 5725 . . . . . . . . . 10 (((1st𝑦) ∈ (mTC‘𝑇) ∧ (𝑔‘(2nd𝑦)) ∈ (mREx‘𝑇)) → ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
3124, 29, 30syl2anc 584 . . . . . . . . 9 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ ∈ ((mTC‘𝑇) × (mREx‘𝑇)))
3231, 21eleqtrrdi 2849 . . . . . . . 8 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ ∈ (mEx‘𝑇))
33 eqidd 2735 . . . . . . . 8 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩))
34 eqidd 2735 . . . . . . . 8 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩))
35 fvex 6919 . . . . . . . . . 10 (1st𝑦) ∈ V
36 fvex 6919 . . . . . . . . . 10 (𝑔‘(2nd𝑦)) ∈ V
3735, 36op1std 8022 . . . . . . . . 9 (𝑥 = ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ → (1st𝑥) = (1st𝑦))
3835, 36op2ndd 8023 . . . . . . . . . 10 (𝑥 = ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ → (2nd𝑥) = (𝑔‘(2nd𝑦)))
3938fveq2d 6910 . . . . . . . . 9 (𝑥 = ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ → (𝑓‘(2nd𝑥)) = (𝑓‘(𝑔‘(2nd𝑦))))
4037, 39opeq12d 4885 . . . . . . . 8 (𝑥 = ⟨(1st𝑦), (𝑔‘(2nd𝑦))⟩ → ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩ = ⟨(1st𝑦), (𝑓‘(𝑔‘(2nd𝑦)))⟩)
4132, 33, 34, 40fmptco 7148 . . . . . . 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 7007 . . . . . . . . . 10 ((𝑔:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ (2nd𝑦) ∈ (mREx‘𝑇)) → ((𝑓𝑔)‘(2nd𝑦)) = (𝑓‘(𝑔‘(2nd𝑦))))
4326, 28, 42syl2anc 584 . . . . . . . . 9 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ((𝑓𝑔)‘(2nd𝑦)) = (𝑓‘(𝑔‘(2nd𝑦))))
4443opeq2d 4884 . . . . . . . 8 (((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) ∧ 𝑦 ∈ (mEx‘𝑇)) → ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩ = ⟨(1st𝑦), (𝑓‘(𝑔‘(2nd𝑦)))⟩)
4544mpteq2dva 5247 . . . . . . 7 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (𝑓‘(𝑔‘(2nd𝑦)))⟩))
4641, 45eqtr4d 2777 . . . . . 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 35505 . . . . . . . 8 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑓𝑔) ∈ ran (mRSubst‘𝑇))
487mptex 7242 . . . . . . . 8 (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩) ∈ V
49 eqid 2734 . . . . . . . . 9 ( ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (‘(2nd𝑦))⟩)) = ( ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (‘(2nd𝑦))⟩))
50 fveq1 6905 . . . . . . . . . . 11 ( = (𝑓𝑔) → (‘(2nd𝑦)) = ((𝑓𝑔)‘(2nd𝑦)))
5150opeq2d 4884 . . . . . . . . . 10 ( = (𝑓𝑔) → ⟨(1st𝑦), (‘(2nd𝑦))⟩ = ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩)
5251mpteq2dv 5249 . . . . . . . . 9 ( = (𝑓𝑔) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (‘(2nd𝑦))⟩) = (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩))
5349, 52elrnmpt1s 5972 . . . . . . . 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 586 . . . . . . 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 35512 . . . . . . 7 ran 𝑆 = ran ( ∈ ran (mRSubst‘𝑇) ↦ (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), (‘(2nd𝑦))⟩))
5654, 55eleqtrrdi 2849 . . . . . 6 ((𝑓 ∈ ran (mRSubst‘𝑇) ∧ 𝑔 ∈ ran (mRSubst‘𝑇)) → (𝑦 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑦), ((𝑓𝑔)‘(2nd𝑦))⟩) ∈ ran 𝑆)
5746, 56eqeltrd 2838 . . . . 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 5870 . . . . . . 7 (𝐹 = (𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) → (𝐹𝐺) = ((𝑥 ∈ (mEx‘𝑇) ↦ ⟨(1st𝑥), (𝑓‘(2nd𝑥))⟩) ∘ 𝐺))
59 coeq2 5871 . . . . . . 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 2794 . . . . . 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 2823 . . . . 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 247 . . . 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 3198 . . 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 235 . 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 598 1 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → (𝐹𝐺) ∈ ran 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wrex 3067  Vcvv 3477  cop 4636  cmpt 5230   × cxp 5686  ran crn 5689  ccom 5692  wf 6558  cfv 6562  1st c1st 8010  2nd c2nd 8011  mTCcmtc 35448  mRExcmrex 35450  mExcmex 35451  mRSubstcmrsub 35454  mSubstcmsub 35455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-word 14549  df-lsw 14597  df-concat 14605  df-s1 14630  df-substr 14675  df-pfx 14705  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-0g 17487  df-gsum 17488  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-frmd 18874  df-vrmd 18875  df-mrex 35470  df-mex 35471  df-mrsub 35474  df-msub 35475
This theorem is referenced by:  mclsppslem  35567
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