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Theorem 1mavmul 21163
 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 2798 . . 3 (.r𝑅) = (.r𝑅)
5 1mavmul.r . . 3 (𝜑𝑅 ∈ Ring)
6 1mavmul.n . . 3 (𝜑𝑁 ∈ Fin)
7 eqid 2798 . . . . 5 (Base‘𝐴) = (Base‘𝐴)
81fveq2i 6649 . . . . 5 (1r𝐴) = (1r‘(𝑁 Mat 𝑅))
91, 7, 8mat1bas 21064 . . . 4 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (1r𝐴) ∈ (Base‘𝐴))
105, 6, 9syl2anc 587 . . 3 (𝜑 → (1r𝐴) ∈ (Base‘𝐴))
11 1mavmul.y . . 3 (𝜑𝑌 ∈ (𝐵m 𝑁))
121, 2, 3, 4, 5, 6, 10, 11mavmulval 21160 . 2 (𝜑 → ((1r𝐴) · 𝑌) = (𝑖𝑁 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖(1r𝐴)𝑗)(.r𝑅)(𝑌𝑗))))))
13 eqid 2798 . . . . . . . . . 10 (1r𝑅) = (1r𝑅)
14 eqid 2798 . . . . . . . . . 10 (0g𝑅) = (0g𝑅)
151, 13, 14mat1 21062 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐴) = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅))))
166, 5, 15syl2anc 587 . . . . . . . 8 (𝜑 → (1r𝐴) = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅))))
1716oveqdr 7164 . . . . . . 7 ((𝜑𝑖𝑁) → (𝑖(1r𝐴)𝑗) = (𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗))
1817oveq1d 7151 . . . . . 6 ((𝜑𝑖𝑁) → ((𝑖(1r𝐴)𝑗)(.r𝑅)(𝑌𝑗)) = ((𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗)(.r𝑅)(𝑌𝑗)))
1918mpteq2dv 5127 . . . . 5 ((𝜑𝑖𝑁) → (𝑗𝑁 ↦ ((𝑖(1r𝐴)𝑗)(.r𝑅)(𝑌𝑗))) = (𝑗𝑁 ↦ ((𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗)(.r𝑅)(𝑌𝑗))))
2019oveq2d 7152 . . . 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 2799 . . . . . . . . 9 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅))) = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅))))
22 eqeq12 2812 . . . . . . . . . . 11 ((𝑥 = 𝑖𝑦 = 𝑗) → (𝑥 = 𝑦𝑖 = 𝑗))
2322ifbid 4447 . . . . . . . . . 10 ((𝑥 = 𝑖𝑦 = 𝑗) → if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
2423adantl 485 . . . . . . . . 9 ((((𝜑𝑖𝑁) ∧ 𝑗𝑁) ∧ (𝑥 = 𝑖𝑦 = 𝑗)) → if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
25 simpr 488 . . . . . . . . . 10 ((𝜑𝑖𝑁) → 𝑖𝑁)
2625adantr 484 . . . . . . . . 9 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → 𝑖𝑁)
27 simpr 488 . . . . . . . . 9 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → 𝑗𝑁)
28 fvexd 6661 . . . . . . . . . 10 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → (1r𝑅) ∈ V)
29 fvexd 6661 . . . . . . . . . 10 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → (0g𝑅) ∈ V)
3028, 29ifcld 4470 . . . . . . . . 9 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) ∈ V)
3121, 24, 26, 27, 30ovmpod 7283 . . . . . . . 8 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → (𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
3231oveq1d 7151 . . . . . . 7 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → ((𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗)(.r𝑅)(𝑌𝑗)) = (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)))
33 iftrue 4431 . . . . . . . . . . . 12 (𝑖 = 𝑗 → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = (1r𝑅))
3433adantr 484 . . . . . . . . . . 11 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = (1r𝑅))
3534oveq1d 7151 . . . . . . . . . 10 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = ((1r𝑅)(.r𝑅)(𝑌𝑗)))
365adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑖𝑁) → 𝑅 ∈ Ring)
3736adantr 484 . . . . . . . . . . . 12 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → 𝑅 ∈ Ring)
383fvexi 6660 . . . . . . . . . . . . . . . . . 18 𝐵 ∈ V
3938a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑𝐵 ∈ V)
4039, 6elmapd 8406 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑌 ∈ (𝐵m 𝑁) ↔ 𝑌:𝑁𝐵))
41 ffvelrn 6827 . . . . . . . . . . . . . . . . 17 ((𝑌:𝑁𝐵𝑗𝑁) → (𝑌𝑗) ∈ 𝐵)
4241ex 416 . . . . . . . . . . . . . . . 16 (𝑌:𝑁𝐵 → (𝑗𝑁 → (𝑌𝑗) ∈ 𝐵))
4340, 42syl6bi 256 . . . . . . . . . . . . . . 15 (𝜑 → (𝑌 ∈ (𝐵m 𝑁) → (𝑗𝑁 → (𝑌𝑗) ∈ 𝐵)))
4411, 43mpd 15 . . . . . . . . . . . . . 14 (𝜑 → (𝑗𝑁 → (𝑌𝑗) ∈ 𝐵))
4544adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑖𝑁) → (𝑗𝑁 → (𝑌𝑗) ∈ 𝐵))
4645imp 410 . . . . . . . . . . . 12 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → (𝑌𝑗) ∈ 𝐵)
473, 4, 13ringlidm 19321 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑌𝑗) ∈ 𝐵) → ((1r𝑅)(.r𝑅)(𝑌𝑗)) = (𝑌𝑗))
4837, 46, 47syl2anc 587 . . . . . . . . . . 11 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → ((1r𝑅)(.r𝑅)(𝑌𝑗)) = (𝑌𝑗))
4948adantl 485 . . . . . . . . . 10 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → ((1r𝑅)(.r𝑅)(𝑌𝑗)) = (𝑌𝑗))
50 fveq2 6646 . . . . . . . . . . . 12 (𝑗 = 𝑖 → (𝑌𝑗) = (𝑌𝑖))
5150equcoms 2027 . . . . . . . . . . 11 (𝑖 = 𝑗 → (𝑌𝑗) = (𝑌𝑖))
5251adantr 484 . . . . . . . . . 10 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (𝑌𝑗) = (𝑌𝑖))
5335, 49, 523eqtrd 2837 . . . . . . . . 9 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = (𝑌𝑖))
54 iftrue 4431 . . . . . . . . . . 11 (𝑗 = 𝑖 → if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)) = (𝑌𝑖))
5554equcoms 2027 . . . . . . . . . 10 (𝑖 = 𝑗 → if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)) = (𝑌𝑖))
5655adantr 484 . . . . . . . . 9 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)) = (𝑌𝑖))
5753, 56eqtr4d 2836 . . . . . . . 8 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
58 iffalse 4434 . . . . . . . . . . 11 𝑖 = 𝑗 → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = (0g𝑅))
5958oveq1d 7151 . . . . . . . . . 10 𝑖 = 𝑗 → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = ((0g𝑅)(.r𝑅)(𝑌𝑗)))
6059adantr 484 . . . . . . . . 9 ((¬ 𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = ((0g𝑅)(.r𝑅)(𝑌𝑗)))
613, 4, 14ringlz 19337 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (𝑌𝑗) ∈ 𝐵) → ((0g𝑅)(.r𝑅)(𝑌𝑗)) = (0g𝑅))
6237, 46, 61syl2anc 587 . . . . . . . . . 10 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → ((0g𝑅)(.r𝑅)(𝑌𝑗)) = (0g𝑅))
6362adantl 485 . . . . . . . . 9 ((¬ 𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → ((0g𝑅)(.r𝑅)(𝑌𝑗)) = (0g𝑅))
64 eqcom 2805 . . . . . . . . . . . 12 (𝑖 = 𝑗𝑗 = 𝑖)
65 iffalse 4434 . . . . . . . . . . . 12 𝑗 = 𝑖 → if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)) = (0g𝑅))
6664, 65sylnbi 333 . . . . . . . . . . 11 𝑖 = 𝑗 → if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)) = (0g𝑅))
6766eqcomd 2804 . . . . . . . . . 10 𝑖 = 𝑗 → (0g𝑅) = if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
6867adantr 484 . . . . . . . . 9 ((¬ 𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (0g𝑅) = if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
6960, 63, 683eqtrd 2837 . . . . . . . 8 ((¬ 𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
7057, 69pm2.61ian 811 . . . . . . 7 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
7132, 70eqtrd 2833 . . . . . 6 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → ((𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗)(.r𝑅)(𝑌𝑗)) = if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
7271mpteq2dva 5126 . . . . 5 ((𝜑𝑖𝑁) → (𝑗𝑁 ↦ ((𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗)(.r𝑅)(𝑌𝑗))) = (𝑗𝑁 ↦ if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅))))
7372oveq2d 7152 . . . 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 19304 . . . . . . 7 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
755, 74syl 17 . . . . . 6 (𝜑𝑅 ∈ Mnd)
7675adantr 484 . . . . 5 ((𝜑𝑖𝑁) → 𝑅 ∈ Mnd)
776adantr 484 . . . . 5 ((𝜑𝑖𝑁) → 𝑁 ∈ Fin)
78 eqid 2798 . . . . 5 (𝑗𝑁 ↦ if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅))) = (𝑗𝑁 ↦ if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
79 ffvelrn 6827 . . . . . . . . . 10 ((𝑌:𝑁𝐵𝑖𝑁) → (𝑌𝑖) ∈ 𝐵)
8079, 3eleqtrdi 2900 . . . . . . . . 9 ((𝑌:𝑁𝐵𝑖𝑁) → (𝑌𝑖) ∈ (Base‘𝑅))
8180ex 416 . . . . . . . 8 (𝑌:𝑁𝐵 → (𝑖𝑁 → (𝑌𝑖) ∈ (Base‘𝑅)))
8240, 81syl6bi 256 . . . . . . 7 (𝜑 → (𝑌 ∈ (𝐵m 𝑁) → (𝑖𝑁 → (𝑌𝑖) ∈ (Base‘𝑅))))
8311, 82mpd 15 . . . . . 6 (𝜑 → (𝑖𝑁 → (𝑌𝑖) ∈ (Base‘𝑅)))
8483imp 410 . . . . 5 ((𝜑𝑖𝑁) → (𝑌𝑖) ∈ (Base‘𝑅))
8514, 76, 77, 25, 78, 84gsummptif1n0 19083 . . . 4 ((𝜑𝑖𝑁) → (𝑅 Σg (𝑗𝑁 ↦ if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))) = (𝑌𝑖))
8620, 73, 853eqtrd 2837 . . 3 ((𝜑𝑖𝑁) → (𝑅 Σg (𝑗𝑁 ↦ ((𝑖(1r𝐴)𝑗)(.r𝑅)(𝑌𝑗)))) = (𝑌𝑖))
8786mpteq2dva 5126 . 2 (𝜑 → (𝑖𝑁 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖(1r𝐴)𝑗)(.r𝑅)(𝑌𝑗))))) = (𝑖𝑁 ↦ (𝑌𝑖)))
88 ffn 6488 . . . . 5 (𝑌:𝑁𝐵𝑌 Fn 𝑁)
8940, 88syl6bi 256 . . . 4 (𝜑 → (𝑌 ∈ (𝐵m 𝑁) → 𝑌 Fn 𝑁))
9011, 89mpd 15 . . 3 (𝜑𝑌 Fn 𝑁)
91 eqcom 2805 . . . 4 ((𝑖𝑁 ↦ (𝑌𝑖)) = 𝑌𝑌 = (𝑖𝑁 ↦ (𝑌𝑖)))
92 dffn5 6700 . . . 4 (𝑌 Fn 𝑁𝑌 = (𝑖𝑁 ↦ (𝑌𝑖)))
9391, 92bitr4i 281 . . 3 ((𝑖𝑁 ↦ (𝑌𝑖)) = 𝑌𝑌 Fn 𝑁)
9490, 93sylibr 237 . 2 (𝜑 → (𝑖𝑁 ↦ (𝑌𝑖)) = 𝑌)
9512, 87, 943eqtrd 2837 1 (𝜑 → ((1r𝐴) · 𝑌) = 𝑌)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ifcif 4425  ⟨cop 4531   ↦ cmpt 5111   Fn wfn 6320  ⟶wf 6321  ‘cfv 6325  (class class class)co 7136   ∈ cmpo 7138   ↑m cmap 8392  Fincfn 8495  Basecbs 16478  .rcmulr 16561  0gc0g 16708   Σg cgsu 16709  Mndcmnd 17906  1rcur 19248  Ringcrg 19294   Mat cmat 21022   maVecMul cmvmul 21155 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-ot 4534  df-uni 4802  df-int 4840  df-iun 4884  df-iin 4885  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-se 5480  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-isom 6334  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-of 7391  df-om 7564  df-1st 7674  df-2nd 7675  df-supp 7817  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-oadd 8092  df-er 8275  df-map 8394  df-ixp 8448  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-fsupp 8821  df-sup 8893  df-oi 8961  df-card 9355  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11629  df-2 11691  df-3 11692  df-4 11693  df-5 11694  df-6 11695  df-7 11696  df-8 11697  df-9 11698  df-n0 11889  df-z 11973  df-dec 12090  df-uz 12235  df-fz 12889  df-fzo 13032  df-seq 13368  df-hash 13690  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-sca 16576  df-vsca 16577  df-ip 16578  df-tset 16579  df-ple 16580  df-ds 16582  df-hom 16584  df-cco 16585  df-0g 16710  df-gsum 16711  df-prds 16716  df-pws 16718  df-mre 16852  df-mrc 16853  df-acs 16855  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-mhm 17951  df-submnd 17952  df-grp 18101  df-minusg 18102  df-sbg 18103  df-mulg 18221  df-subg 18272  df-ghm 18352  df-cntz 18443  df-cmn 18904  df-abl 18905  df-mgp 19237  df-ur 19249  df-ring 19296  df-subrg 19530  df-lmod 19633  df-lss 19701  df-sra 19941  df-rgmod 19942  df-dsmm 20426  df-frlm 20441  df-mamu 21001  df-mat 21023  df-mvmul 21156 This theorem is referenced by:  slesolinv  21295  slesolinvbi  21296
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