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Theorem 1mavmul 20841
Description: Multiplication of the identity NxN matrix with an N-dimensional vector results in the vector itself. (Contributed by AV, 9-Feb-2019.) (Revised by AV, 23-Feb-2019.)
Hypotheses
Ref Expression
1mavmul.a 𝐴 = (𝑁 Mat 𝑅)
1mavmul.b 𝐵 = (Base‘𝑅)
1mavmul.t · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)
1mavmul.r (𝜑𝑅 ∈ Ring)
1mavmul.n (𝜑𝑁 ∈ Fin)
1mavmul.y (𝜑𝑌 ∈ (𝐵𝑚 𝑁))
Assertion
Ref Expression
1mavmul (𝜑 → ((1r𝐴) · 𝑌) = 𝑌)

Proof of Theorem 1mavmul
Dummy variables 𝑖 𝑗 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1mavmul.a . . 3 𝐴 = (𝑁 Mat 𝑅)
2 1mavmul.t . . 3 · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)
3 1mavmul.b . . 3 𝐵 = (Base‘𝑅)
4 eqid 2795 . . 3 (.r𝑅) = (.r𝑅)
5 1mavmul.r . . 3 (𝜑𝑅 ∈ Ring)
6 1mavmul.n . . 3 (𝜑𝑁 ∈ Fin)
7 eqid 2795 . . . . 5 (Base‘𝐴) = (Base‘𝐴)
81fveq2i 6541 . . . . 5 (1r𝐴) = (1r‘(𝑁 Mat 𝑅))
91, 7, 8mat1bas 20742 . . . 4 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (1r𝐴) ∈ (Base‘𝐴))
105, 6, 9syl2anc 584 . . 3 (𝜑 → (1r𝐴) ∈ (Base‘𝐴))
11 1mavmul.y . . 3 (𝜑𝑌 ∈ (𝐵𝑚 𝑁))
121, 2, 3, 4, 5, 6, 10, 11mavmulval 20838 . 2 (𝜑 → ((1r𝐴) · 𝑌) = (𝑖𝑁 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖(1r𝐴)𝑗)(.r𝑅)(𝑌𝑗))))))
13 eqid 2795 . . . . . . . . . 10 (1r𝑅) = (1r𝑅)
14 eqid 2795 . . . . . . . . . 10 (0g𝑅) = (0g𝑅)
151, 13, 14mat1 20740 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐴) = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅))))
166, 5, 15syl2anc 584 . . . . . . . 8 (𝜑 → (1r𝐴) = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅))))
1716oveqdr 7044 . . . . . . 7 ((𝜑𝑖𝑁) → (𝑖(1r𝐴)𝑗) = (𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗))
1817oveq1d 7031 . . . . . 6 ((𝜑𝑖𝑁) → ((𝑖(1r𝐴)𝑗)(.r𝑅)(𝑌𝑗)) = ((𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗)(.r𝑅)(𝑌𝑗)))
1918mpteq2dv 5056 . . . . 5 ((𝜑𝑖𝑁) → (𝑗𝑁 ↦ ((𝑖(1r𝐴)𝑗)(.r𝑅)(𝑌𝑗))) = (𝑗𝑁 ↦ ((𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗)(.r𝑅)(𝑌𝑗))))
2019oveq2d 7032 . . . 4 ((𝜑𝑖𝑁) → (𝑅 Σg (𝑗𝑁 ↦ ((𝑖(1r𝐴)𝑗)(.r𝑅)(𝑌𝑗)))) = (𝑅 Σg (𝑗𝑁 ↦ ((𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗)(.r𝑅)(𝑌𝑗)))))
21 eqidd 2796 . . . . . . . . 9 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅))) = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅))))
22 eqeq12 2808 . . . . . . . . . . 11 ((𝑥 = 𝑖𝑦 = 𝑗) → (𝑥 = 𝑦𝑖 = 𝑗))
2322ifbid 4403 . . . . . . . . . 10 ((𝑥 = 𝑖𝑦 = 𝑗) → if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
2423adantl 482 . . . . . . . . 9 ((((𝜑𝑖𝑁) ∧ 𝑗𝑁) ∧ (𝑥 = 𝑖𝑦 = 𝑗)) → if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
25 simpr 485 . . . . . . . . . 10 ((𝜑𝑖𝑁) → 𝑖𝑁)
2625adantr 481 . . . . . . . . 9 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → 𝑖𝑁)
27 simpr 485 . . . . . . . . 9 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → 𝑗𝑁)
28 fvexd 6553 . . . . . . . . . 10 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → (1r𝑅) ∈ V)
29 fvexd 6553 . . . . . . . . . 10 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → (0g𝑅) ∈ V)
3028, 29ifcld 4426 . . . . . . . . 9 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) ∈ V)
3121, 24, 26, 27, 30ovmpod 7158 . . . . . . . 8 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → (𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
3231oveq1d 7031 . . . . . . 7 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → ((𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗)(.r𝑅)(𝑌𝑗)) = (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)))
33 iftrue 4387 . . . . . . . . . . . 12 (𝑖 = 𝑗 → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = (1r𝑅))
3433adantr 481 . . . . . . . . . . 11 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = (1r𝑅))
3534oveq1d 7031 . . . . . . . . . 10 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = ((1r𝑅)(.r𝑅)(𝑌𝑗)))
365adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑖𝑁) → 𝑅 ∈ Ring)
3736adantr 481 . . . . . . . . . . . 12 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → 𝑅 ∈ Ring)
383fvexi 6552 . . . . . . . . . . . . . . . . . 18 𝐵 ∈ V
3938a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑𝐵 ∈ V)
4039, 6elmapd 8270 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑌 ∈ (𝐵𝑚 𝑁) ↔ 𝑌:𝑁𝐵))
41 ffvelrn 6714 . . . . . . . . . . . . . . . . 17 ((𝑌:𝑁𝐵𝑗𝑁) → (𝑌𝑗) ∈ 𝐵)
4241ex 413 . . . . . . . . . . . . . . . 16 (𝑌:𝑁𝐵 → (𝑗𝑁 → (𝑌𝑗) ∈ 𝐵))
4340, 42syl6bi 254 . . . . . . . . . . . . . . 15 (𝜑 → (𝑌 ∈ (𝐵𝑚 𝑁) → (𝑗𝑁 → (𝑌𝑗) ∈ 𝐵)))
4411, 43mpd 15 . . . . . . . . . . . . . 14 (𝜑 → (𝑗𝑁 → (𝑌𝑗) ∈ 𝐵))
4544adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑖𝑁) → (𝑗𝑁 → (𝑌𝑗) ∈ 𝐵))
4645imp 407 . . . . . . . . . . . 12 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → (𝑌𝑗) ∈ 𝐵)
473, 4, 13ringlidm 19011 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑌𝑗) ∈ 𝐵) → ((1r𝑅)(.r𝑅)(𝑌𝑗)) = (𝑌𝑗))
4837, 46, 47syl2anc 584 . . . . . . . . . . 11 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → ((1r𝑅)(.r𝑅)(𝑌𝑗)) = (𝑌𝑗))
4948adantl 482 . . . . . . . . . 10 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → ((1r𝑅)(.r𝑅)(𝑌𝑗)) = (𝑌𝑗))
50 fveq2 6538 . . . . . . . . . . . 12 (𝑗 = 𝑖 → (𝑌𝑗) = (𝑌𝑖))
5150equcoms 2004 . . . . . . . . . . 11 (𝑖 = 𝑗 → (𝑌𝑗) = (𝑌𝑖))
5251adantr 481 . . . . . . . . . 10 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (𝑌𝑗) = (𝑌𝑖))
5335, 49, 523eqtrd 2835 . . . . . . . . 9 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = (𝑌𝑖))
54 iftrue 4387 . . . . . . . . . . 11 (𝑗 = 𝑖 → if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)) = (𝑌𝑖))
5554equcoms 2004 . . . . . . . . . 10 (𝑖 = 𝑗 → if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)) = (𝑌𝑖))
5655adantr 481 . . . . . . . . 9 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)) = (𝑌𝑖))
5753, 56eqtr4d 2834 . . . . . . . 8 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
58 iffalse 4390 . . . . . . . . . . 11 𝑖 = 𝑗 → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = (0g𝑅))
5958oveq1d 7031 . . . . . . . . . 10 𝑖 = 𝑗 → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = ((0g𝑅)(.r𝑅)(𝑌𝑗)))
6059adantr 481 . . . . . . . . 9 ((¬ 𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = ((0g𝑅)(.r𝑅)(𝑌𝑗)))
613, 4, 14ringlz 19027 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (𝑌𝑗) ∈ 𝐵) → ((0g𝑅)(.r𝑅)(𝑌𝑗)) = (0g𝑅))
6237, 46, 61syl2anc 584 . . . . . . . . . 10 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → ((0g𝑅)(.r𝑅)(𝑌𝑗)) = (0g𝑅))
6362adantl 482 . . . . . . . . 9 ((¬ 𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → ((0g𝑅)(.r𝑅)(𝑌𝑗)) = (0g𝑅))
64 eqcom 2802 . . . . . . . . . . . 12 (𝑖 = 𝑗𝑗 = 𝑖)
65 iffalse 4390 . . . . . . . . . . . 12 𝑗 = 𝑖 → if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)) = (0g𝑅))
6664, 65sylnbi 331 . . . . . . . . . . 11 𝑖 = 𝑗 → if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)) = (0g𝑅))
6766eqcomd 2801 . . . . . . . . . 10 𝑖 = 𝑗 → (0g𝑅) = if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
6867adantr 481 . . . . . . . . 9 ((¬ 𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (0g𝑅) = if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
6960, 63, 683eqtrd 2835 . . . . . . . 8 ((¬ 𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
7057, 69pm2.61ian 808 . . . . . . 7 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
7132, 70eqtrd 2831 . . . . . 6 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → ((𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗)(.r𝑅)(𝑌𝑗)) = if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
7271mpteq2dva 5055 . . . . 5 ((𝜑𝑖𝑁) → (𝑗𝑁 ↦ ((𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗)(.r𝑅)(𝑌𝑗))) = (𝑗𝑁 ↦ if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅))))
7372oveq2d 7032 . . . 4 ((𝜑𝑖𝑁) → (𝑅 Σg (𝑗𝑁 ↦ ((𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗)(.r𝑅)(𝑌𝑗)))) = (𝑅 Σg (𝑗𝑁 ↦ if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))))
74 ringmnd 18996 . . . . . . 7 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
755, 74syl 17 . . . . . 6 (𝜑𝑅 ∈ Mnd)
7675adantr 481 . . . . 5 ((𝜑𝑖𝑁) → 𝑅 ∈ Mnd)
776adantr 481 . . . . 5 ((𝜑𝑖𝑁) → 𝑁 ∈ Fin)
78 eqid 2795 . . . . 5 (𝑗𝑁 ↦ if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅))) = (𝑗𝑁 ↦ if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
79 ffvelrn 6714 . . . . . . . . . 10 ((𝑌:𝑁𝐵𝑖𝑁) → (𝑌𝑖) ∈ 𝐵)
8079, 3syl6eleq 2893 . . . . . . . . 9 ((𝑌:𝑁𝐵𝑖𝑁) → (𝑌𝑖) ∈ (Base‘𝑅))
8180ex 413 . . . . . . . 8 (𝑌:𝑁𝐵 → (𝑖𝑁 → (𝑌𝑖) ∈ (Base‘𝑅)))
8240, 81syl6bi 254 . . . . . . 7 (𝜑 → (𝑌 ∈ (𝐵𝑚 𝑁) → (𝑖𝑁 → (𝑌𝑖) ∈ (Base‘𝑅))))
8311, 82mpd 15 . . . . . 6 (𝜑 → (𝑖𝑁 → (𝑌𝑖) ∈ (Base‘𝑅)))
8483imp 407 . . . . 5 ((𝜑𝑖𝑁) → (𝑌𝑖) ∈ (Base‘𝑅))
8514, 76, 77, 25, 78, 84gsummptif1n0 18806 . . . 4 ((𝜑𝑖𝑁) → (𝑅 Σg (𝑗𝑁 ↦ if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))) = (𝑌𝑖))
8620, 73, 853eqtrd 2835 . . 3 ((𝜑𝑖𝑁) → (𝑅 Σg (𝑗𝑁 ↦ ((𝑖(1r𝐴)𝑗)(.r𝑅)(𝑌𝑗)))) = (𝑌𝑖))
8786mpteq2dva 5055 . 2 (𝜑 → (𝑖𝑁 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖(1r𝐴)𝑗)(.r𝑅)(𝑌𝑗))))) = (𝑖𝑁 ↦ (𝑌𝑖)))
88 ffn 6382 . . . . 5 (𝑌:𝑁𝐵𝑌 Fn 𝑁)
8940, 88syl6bi 254 . . . 4 (𝜑 → (𝑌 ∈ (𝐵𝑚 𝑁) → 𝑌 Fn 𝑁))
9011, 89mpd 15 . . 3 (𝜑𝑌 Fn 𝑁)
91 eqcom 2802 . . . 4 ((𝑖𝑁 ↦ (𝑌𝑖)) = 𝑌𝑌 = (𝑖𝑁 ↦ (𝑌𝑖)))
92 dffn5 6592 . . . 4 (𝑌 Fn 𝑁𝑌 = (𝑖𝑁 ↦ (𝑌𝑖)))
9391, 92bitr4i 279 . . 3 ((𝑖𝑁 ↦ (𝑌𝑖)) = 𝑌𝑌 Fn 𝑁)
9490, 93sylibr 235 . 2 (𝜑 → (𝑖𝑁 ↦ (𝑌𝑖)) = 𝑌)
9512, 87, 943eqtrd 2835 1 (𝜑 → ((1r𝐴) · 𝑌) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1522  wcel 2081  Vcvv 3437  ifcif 4381  cop 4478  cmpt 5041   Fn wfn 6220  wf 6221  cfv 6225  (class class class)co 7016  cmpo 7018  𝑚 cmap 8256  Fincfn 8357  Basecbs 16312  .rcmulr 16395  0gc0g 16542   Σg cgsu 16543  Mndcmnd 17733  1rcur 18941  Ringcrg 18987   Mat cmat 20700   maVecMul cmvmul 20833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-ot 4481  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-of 7267  df-om 7437  df-1st 7545  df-2nd 7546  df-supp 7682  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-map 8258  df-ixp 8311  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-fsupp 8680  df-sup 8752  df-oi 8820  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-3 11549  df-4 11550  df-5 11551  df-6 11552  df-7 11553  df-8 11554  df-9 11555  df-n0 11746  df-z 11830  df-dec 11948  df-uz 12094  df-fz 12743  df-fzo 12884  df-seq 13220  df-hash 13541  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-ress 16320  df-plusg 16407  df-mulr 16408  df-sca 16410  df-vsca 16411  df-ip 16412  df-tset 16413  df-ple 16414  df-ds 16416  df-hom 16418  df-cco 16419  df-0g 16544  df-gsum 16545  df-prds 16550  df-pws 16552  df-mre 16686  df-mrc 16687  df-acs 16689  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-mhm 17774  df-submnd 17775  df-grp 17864  df-minusg 17865  df-sbg 17866  df-mulg 17982  df-subg 18030  df-ghm 18097  df-cntz 18188  df-cmn 18635  df-abl 18636  df-mgp 18930  df-ur 18942  df-ring 18989  df-subrg 19223  df-lmod 19326  df-lss 19394  df-sra 19634  df-rgmod 19635  df-dsmm 20558  df-frlm 20573  df-mamu 20677  df-mat 20701  df-mvmul 20834
This theorem is referenced by:  slesolinv  20973  slesolinvbi  20974
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