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Theorem scmatf1 20251
Description: There is a 1-1 function from a ring to any ring of scalar matrices with positive dimension over this ring. (Contributed by AV, 25-Dec-2019.)
Hypotheses
Ref Expression
scmatrhmval.k 𝐾 = (Base‘𝑅)
scmatrhmval.a 𝐴 = (𝑁 Mat 𝑅)
scmatrhmval.o 1 = (1r𝐴)
scmatrhmval.t = ( ·𝑠𝐴)
scmatrhmval.f 𝐹 = (𝑥𝐾 ↦ (𝑥 1 ))
scmatrhmval.c 𝐶 = (𝑁 ScMat 𝑅)
Assertion
Ref Expression
scmatf1 ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝐹:𝐾1-1𝐶)
Distinct variable groups:   𝑥,𝐾   𝑥,𝑅   𝑥, 1   𝑥,   𝑥,𝐶   𝑥,𝑁
Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)

Proof of Theorem scmatf1
Dummy variables 𝑦 𝑧 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 scmatrhmval.k . . . 4 𝐾 = (Base‘𝑅)
2 scmatrhmval.a . . . 4 𝐴 = (𝑁 Mat 𝑅)
3 scmatrhmval.o . . . 4 1 = (1r𝐴)
4 scmatrhmval.t . . . 4 = ( ·𝑠𝐴)
5 scmatrhmval.f . . . 4 𝐹 = (𝑥𝐾 ↦ (𝑥 1 ))
6 scmatrhmval.c . . . 4 𝐶 = (𝑁 ScMat 𝑅)
71, 2, 3, 4, 5, 6scmatf 20249 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹:𝐾𝐶)
873adant2 1078 . 2 ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝐹:𝐾𝐶)
9 simpr 477 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
10 simpl 473 . . . . . . 7 ((𝑦𝐾𝑧𝐾) → 𝑦𝐾)
111, 2, 3, 4, 5scmatrhmval 20247 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑦𝐾) → (𝐹𝑦) = (𝑦 1 ))
129, 10, 11syl2an 494 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → (𝐹𝑦) = (𝑦 1 ))
13 simpr 477 . . . . . . 7 ((𝑦𝐾𝑧𝐾) → 𝑧𝐾)
141, 2, 3, 4, 5scmatrhmval 20247 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑧𝐾) → (𝐹𝑧) = (𝑧 1 ))
159, 13, 14syl2an 494 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → (𝐹𝑧) = (𝑧 1 ))
1612, 15eqeq12d 2641 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → ((𝐹𝑦) = (𝐹𝑧) ↔ (𝑦 1 ) = (𝑧 1 )))
17163adantl2 1216 . . . 4 (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → ((𝐹𝑦) = (𝐹𝑧) ↔ (𝑦 1 ) = (𝑧 1 )))
182matring 20163 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
19 eqid 2626 . . . . . . . . . . . 12 (Base‘𝐴) = (Base‘𝐴)
2019, 3ringidcl 18484 . . . . . . . . . . 11 (𝐴 ∈ Ring → 1 ∈ (Base‘𝐴))
2118, 20syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 ∈ (Base‘𝐴))
2221, 10anim12ci 590 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → (𝑦𝐾1 ∈ (Base‘𝐴)))
231, 2, 19, 4matvscl 20151 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾1 ∈ (Base‘𝐴))) → (𝑦 1 ) ∈ (Base‘𝐴))
2422, 23syldan 487 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → (𝑦 1 ) ∈ (Base‘𝐴))
2521, 13anim12ci 590 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → (𝑧𝐾1 ∈ (Base‘𝐴)))
261, 2, 19, 4matvscl 20151 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑧𝐾1 ∈ (Base‘𝐴))) → (𝑧 1 ) ∈ (Base‘𝐴))
2725, 26syldan 487 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → (𝑧 1 ) ∈ (Base‘𝐴))
2824, 27jca 554 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → ((𝑦 1 ) ∈ (Base‘𝐴) ∧ (𝑧 1 ) ∈ (Base‘𝐴)))
29283adantl2 1216 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → ((𝑦 1 ) ∈ (Base‘𝐴) ∧ (𝑧 1 ) ∈ (Base‘𝐴)))
302, 19eqmat 20144 . . . . . 6 (((𝑦 1 ) ∈ (Base‘𝐴) ∧ (𝑧 1 ) ∈ (Base‘𝐴)) → ((𝑦 1 ) = (𝑧 1 ) ↔ ∀𝑖𝑁𝑗𝑁 (𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗)))
3129, 30syl 17 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → ((𝑦 1 ) = (𝑧 1 ) ↔ ∀𝑖𝑁𝑗𝑁 (𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗)))
32 difsnid 4315 . . . . . . . . . . . 12 (𝑖𝑁 → ((𝑁 ∖ {𝑖}) ∪ {𝑖}) = 𝑁)
3332eqcomd 2632 . . . . . . . . . . 11 (𝑖𝑁𝑁 = ((𝑁 ∖ {𝑖}) ∪ {𝑖}))
3433adantl 482 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) ∧ 𝑖𝑁) → 𝑁 = ((𝑁 ∖ {𝑖}) ∪ {𝑖}))
3534raleqdv 3138 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) ∧ 𝑖𝑁) → (∀𝑗𝑁 (𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗) ↔ ∀𝑗 ∈ ((𝑁 ∖ {𝑖}) ∪ {𝑖})(𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗)))
36 oveq2 6613 . . . . . . . . . . . 12 (𝑗 = 𝑖 → (𝑖(𝑦 1 )𝑗) = (𝑖(𝑦 1 )𝑖))
37 oveq2 6613 . . . . . . . . . . . 12 (𝑗 = 𝑖 → (𝑖(𝑧 1 )𝑗) = (𝑖(𝑧 1 )𝑖))
3836, 37eqeq12d 2641 . . . . . . . . . . 11 (𝑗 = 𝑖 → ((𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗) ↔ (𝑖(𝑦 1 )𝑖) = (𝑖(𝑧 1 )𝑖)))
3938ralunsn 4395 . . . . . . . . . 10 (𝑖𝑁 → (∀𝑗 ∈ ((𝑁 ∖ {𝑖}) ∪ {𝑖})(𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗) ↔ (∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗) ∧ (𝑖(𝑦 1 )𝑖) = (𝑖(𝑧 1 )𝑖))))
4039adantl 482 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) ∧ 𝑖𝑁) → (∀𝑗 ∈ ((𝑁 ∖ {𝑖}) ∪ {𝑖})(𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗) ↔ (∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗) ∧ (𝑖(𝑦 1 )𝑖) = (𝑖(𝑧 1 )𝑖))))
4110anim2i 592 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦𝐾))
42 df-3an 1038 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐾) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦𝐾))
4341, 42sylibr 224 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐾))
44 id 22 . . . . . . . . . . . . . 14 (𝑖𝑁𝑖𝑁)
4544, 44jca 554 . . . . . . . . . . . . 13 (𝑖𝑁 → (𝑖𝑁𝑖𝑁))
46 eqid 2626 . . . . . . . . . . . . . 14 (0g𝑅) = (0g𝑅)
472, 1, 46, 3, 4scmatscmide 20227 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦𝐾) ∧ (𝑖𝑁𝑖𝑁)) → (𝑖(𝑦 1 )𝑖) = if(𝑖 = 𝑖, 𝑦, (0g𝑅)))
4843, 45, 47syl2an 494 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) ∧ 𝑖𝑁) → (𝑖(𝑦 1 )𝑖) = if(𝑖 = 𝑖, 𝑦, (0g𝑅)))
49 eqid 2626 . . . . . . . . . . . . 13 𝑖 = 𝑖
5049iftruei 4070 . . . . . . . . . . . 12 if(𝑖 = 𝑖, 𝑦, (0g𝑅)) = 𝑦
5148, 50syl6eq 2676 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) ∧ 𝑖𝑁) → (𝑖(𝑦 1 )𝑖) = 𝑦)
5213anim2i 592 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑧𝐾))
53 df-3an 1038 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑧𝐾) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑧𝐾))
5452, 53sylibr 224 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑧𝐾))
552, 1, 46, 3, 4scmatscmide 20227 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑧𝐾) ∧ (𝑖𝑁𝑖𝑁)) → (𝑖(𝑧 1 )𝑖) = if(𝑖 = 𝑖, 𝑧, (0g𝑅)))
5654, 45, 55syl2an 494 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) ∧ 𝑖𝑁) → (𝑖(𝑧 1 )𝑖) = if(𝑖 = 𝑖, 𝑧, (0g𝑅)))
5749iftruei 4070 . . . . . . . . . . . 12 if(𝑖 = 𝑖, 𝑧, (0g𝑅)) = 𝑧
5856, 57syl6eq 2676 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) ∧ 𝑖𝑁) → (𝑖(𝑧 1 )𝑖) = 𝑧)
5951, 58eqeq12d 2641 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) ∧ 𝑖𝑁) → ((𝑖(𝑦 1 )𝑖) = (𝑖(𝑧 1 )𝑖) ↔ 𝑦 = 𝑧))
6059anbi2d 739 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) ∧ 𝑖𝑁) → ((∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗) ∧ (𝑖(𝑦 1 )𝑖) = (𝑖(𝑧 1 )𝑖)) ↔ (∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗) ∧ 𝑦 = 𝑧)))
6135, 40, 603bitrd 294 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) ∧ 𝑖𝑁) → (∀𝑗𝑁 (𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗) ↔ (∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗) ∧ 𝑦 = 𝑧)))
6261ralbidva 2984 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → (∀𝑖𝑁𝑗𝑁 (𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗) ↔ ∀𝑖𝑁 (∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗) ∧ 𝑦 = 𝑧)))
63623adantl2 1216 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → (∀𝑖𝑁𝑗𝑁 (𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗) ↔ ∀𝑖𝑁 (∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗) ∧ 𝑦 = 𝑧)))
64 r19.26 3062 . . . . . . . 8 (∀𝑖𝑁 (∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗) ∧ 𝑦 = 𝑧) ↔ (∀𝑖𝑁𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗) ∧ ∀𝑖𝑁 𝑦 = 𝑧))
65 rspn0 3915 . . . . . . . . . . . 12 (𝑁 ≠ ∅ → (∀𝑖𝑁 𝑦 = 𝑧𝑦 = 𝑧))
66653ad2ant2 1081 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → (∀𝑖𝑁 𝑦 = 𝑧𝑦 = 𝑧))
6766adantr 481 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → (∀𝑖𝑁 𝑦 = 𝑧𝑦 = 𝑧))
6867com12 32 . . . . . . . . 9 (∀𝑖𝑁 𝑦 = 𝑧 → (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → 𝑦 = 𝑧))
6968adantl 482 . . . . . . . 8 ((∀𝑖𝑁𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗) ∧ ∀𝑖𝑁 𝑦 = 𝑧) → (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → 𝑦 = 𝑧))
7064, 69sylbi 207 . . . . . . 7 (∀𝑖𝑁 (∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗) ∧ 𝑦 = 𝑧) → (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → 𝑦 = 𝑧))
7170com12 32 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → (∀𝑖𝑁 (∀𝑗 ∈ (𝑁 ∖ {𝑖})(𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗) ∧ 𝑦 = 𝑧) → 𝑦 = 𝑧))
7263, 71sylbid 230 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → (∀𝑖𝑁𝑗𝑁 (𝑖(𝑦 1 )𝑗) = (𝑖(𝑧 1 )𝑗) → 𝑦 = 𝑧))
7331, 72sylbid 230 . . . 4 (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → ((𝑦 1 ) = (𝑧 1 ) → 𝑦 = 𝑧))
7417, 73sylbid 230 . . 3 (((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑦𝐾𝑧𝐾)) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
7574ralrimivva 2970 . 2 ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → ∀𝑦𝐾𝑧𝐾 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
76 dff13 6467 . 2 (𝐹:𝐾1-1𝐶 ↔ (𝐹:𝐾𝐶 ∧ ∀𝑦𝐾𝑧𝐾 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧)))
778, 75, 76sylanbrc 697 1 ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝐹:𝐾1-1𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796  wral 2912  cdif 3557  cun 3558  c0 3896  ifcif 4063  {csn 4153  cmpt 4678  wf 5846  1-1wf1 5847  cfv 5850  (class class class)co 6605  Fincfn 7900  Basecbs 15776   ·𝑠 cvsca 15861  0gc0g 16016  1rcur 18417  Ringcrg 18463   Mat cmat 20127   ScMat cscmat 20209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-ot 4162  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-of 6851  df-om 7014  df-1st 7116  df-2nd 7117  df-supp 7242  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-map 7805  df-ixp 7854  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-fsupp 8221  df-sup 8293  df-oi 8360  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-6 11028  df-7 11029  df-8 11030  df-9 11031  df-n0 11238  df-z 11323  df-dec 11438  df-uz 11632  df-fz 12266  df-fzo 12404  df-seq 12739  df-hash 13055  df-struct 15778  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-mulr 15871  df-sca 15873  df-vsca 15874  df-ip 15875  df-tset 15876  df-ple 15877  df-ds 15880  df-hom 15882  df-cco 15883  df-0g 16018  df-gsum 16019  df-prds 16024  df-pws 16026  df-mre 16162  df-mrc 16163  df-acs 16165  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-mhm 17251  df-submnd 17252  df-grp 17341  df-minusg 17342  df-sbg 17343  df-mulg 17457  df-subg 17507  df-ghm 17574  df-cntz 17666  df-cmn 18111  df-abl 18112  df-mgp 18406  df-ur 18418  df-ring 18465  df-subrg 18694  df-lmod 18781  df-lss 18847  df-sra 19086  df-rgmod 19087  df-dsmm 19990  df-frlm 20005  df-mamu 20104  df-mat 20128  df-scmat 20211
This theorem is referenced by:  scmatf1o  20252
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