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Theorem 1mavmul 20402
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 2651 . . 3 (.r𝑅) = (.r𝑅)
5 1mavmul.r . . 3 (𝜑𝑅 ∈ Ring)
6 1mavmul.n . . 3 (𝜑𝑁 ∈ Fin)
7 eqid 2651 . . . . 5 (Base‘𝐴) = (Base‘𝐴)
81fveq2i 6232 . . . . 5 (1r𝐴) = (1r‘(𝑁 Mat 𝑅))
91, 7, 8mat1bas 20303 . . . 4 ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (1r𝐴) ∈ (Base‘𝐴))
105, 6, 9syl2anc 694 . . 3 (𝜑 → (1r𝐴) ∈ (Base‘𝐴))
11 1mavmul.y . . 3 (𝜑𝑌 ∈ (𝐵𝑚 𝑁))
121, 2, 3, 4, 5, 6, 10, 11mavmulval 20399 . 2 (𝜑 → ((1r𝐴) · 𝑌) = (𝑖𝑁 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖(1r𝐴)𝑗)(.r𝑅)(𝑌𝑗))))))
13 eqid 2651 . . . . . . . . . 10 (1r𝑅) = (1r𝑅)
14 eqid 2651 . . . . . . . . . 10 (0g𝑅) = (0g𝑅)
151, 13, 14mat1 20301 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐴) = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅))))
166, 5, 15syl2anc 694 . . . . . . . 8 (𝜑 → (1r𝐴) = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅))))
1716oveqdr 6714 . . . . . . 7 ((𝜑𝑖𝑁) → (𝑖(1r𝐴)𝑗) = (𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗))
1817oveq1d 6705 . . . . . 6 ((𝜑𝑖𝑁) → ((𝑖(1r𝐴)𝑗)(.r𝑅)(𝑌𝑗)) = ((𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗)(.r𝑅)(𝑌𝑗)))
1918mpteq2dv 4778 . . . . 5 ((𝜑𝑖𝑁) → (𝑗𝑁 ↦ ((𝑖(1r𝐴)𝑗)(.r𝑅)(𝑌𝑗))) = (𝑗𝑁 ↦ ((𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗)(.r𝑅)(𝑌𝑗))))
2019oveq2d 6706 . . . 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 2652 . . . . . . . . 9 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅))) = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅))))
22 eqeq12 2664 . . . . . . . . . . 11 ((𝑥 = 𝑖𝑦 = 𝑗) → (𝑥 = 𝑦𝑖 = 𝑗))
2322ifbid 4141 . . . . . . . . . 10 ((𝑥 = 𝑖𝑦 = 𝑗) → if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
2423adantl 481 . . . . . . . . 9 ((((𝜑𝑖𝑁) ∧ 𝑗𝑁) ∧ (𝑥 = 𝑖𝑦 = 𝑗)) → if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
25 simpr 476 . . . . . . . . . 10 ((𝜑𝑖𝑁) → 𝑖𝑁)
2625adantr 480 . . . . . . . . 9 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → 𝑖𝑁)
27 simpr 476 . . . . . . . . 9 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → 𝑗𝑁)
28 fvexd 6241 . . . . . . . . . 10 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → (1r𝑅) ∈ V)
29 fvexd 6241 . . . . . . . . . 10 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → (0g𝑅) ∈ V)
3028, 29ifcld 4164 . . . . . . . . 9 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) ∈ V)
3121, 24, 26, 27, 30ovmpt2d 6830 . . . . . . . 8 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → (𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
3231oveq1d 6705 . . . . . . 7 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → ((𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗)(.r𝑅)(𝑌𝑗)) = (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)))
33 iftrue 4125 . . . . . . . . . . . 12 (𝑖 = 𝑗 → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = (1r𝑅))
3433adantr 480 . . . . . . . . . . 11 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = (1r𝑅))
3534oveq1d 6705 . . . . . . . . . 10 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = ((1r𝑅)(.r𝑅)(𝑌𝑗)))
365adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑖𝑁) → 𝑅 ∈ Ring)
3736adantr 480 . . . . . . . . . . . 12 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → 𝑅 ∈ Ring)
38 fvex 6239 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑅) ∈ V
393, 38eqeltri 2726 . . . . . . . . . . . . . . . . . 18 𝐵 ∈ V
4039a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑𝐵 ∈ V)
4140, 6elmapd 7913 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑌 ∈ (𝐵𝑚 𝑁) ↔ 𝑌:𝑁𝐵))
42 ffvelrn 6397 . . . . . . . . . . . . . . . . 17 ((𝑌:𝑁𝐵𝑗𝑁) → (𝑌𝑗) ∈ 𝐵)
4342ex 449 . . . . . . . . . . . . . . . 16 (𝑌:𝑁𝐵 → (𝑗𝑁 → (𝑌𝑗) ∈ 𝐵))
4441, 43syl6bi 243 . . . . . . . . . . . . . . 15 (𝜑 → (𝑌 ∈ (𝐵𝑚 𝑁) → (𝑗𝑁 → (𝑌𝑗) ∈ 𝐵)))
4511, 44mpd 15 . . . . . . . . . . . . . 14 (𝜑 → (𝑗𝑁 → (𝑌𝑗) ∈ 𝐵))
4645adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑖𝑁) → (𝑗𝑁 → (𝑌𝑗) ∈ 𝐵))
4746imp 444 . . . . . . . . . . . 12 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → (𝑌𝑗) ∈ 𝐵)
483, 4, 13ringlidm 18617 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑌𝑗) ∈ 𝐵) → ((1r𝑅)(.r𝑅)(𝑌𝑗)) = (𝑌𝑗))
4937, 47, 48syl2anc 694 . . . . . . . . . . 11 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → ((1r𝑅)(.r𝑅)(𝑌𝑗)) = (𝑌𝑗))
5049adantl 481 . . . . . . . . . 10 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → ((1r𝑅)(.r𝑅)(𝑌𝑗)) = (𝑌𝑗))
51 fveq2 6229 . . . . . . . . . . . 12 (𝑗 = 𝑖 → (𝑌𝑗) = (𝑌𝑖))
5251equcoms 1993 . . . . . . . . . . 11 (𝑖 = 𝑗 → (𝑌𝑗) = (𝑌𝑖))
5352adantr 480 . . . . . . . . . 10 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (𝑌𝑗) = (𝑌𝑖))
5435, 50, 533eqtrd 2689 . . . . . . . . 9 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = (𝑌𝑖))
55 iftrue 4125 . . . . . . . . . . 11 (𝑗 = 𝑖 → if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)) = (𝑌𝑖))
5655equcoms 1993 . . . . . . . . . 10 (𝑖 = 𝑗 → if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)) = (𝑌𝑖))
5756adantr 480 . . . . . . . . 9 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)) = (𝑌𝑖))
5854, 57eqtr4d 2688 . . . . . . . 8 ((𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
59 iffalse 4128 . . . . . . . . . . 11 𝑖 = 𝑗 → if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)) = (0g𝑅))
6059oveq1d 6705 . . . . . . . . . 10 𝑖 = 𝑗 → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = ((0g𝑅)(.r𝑅)(𝑌𝑗)))
6160adantr 480 . . . . . . . . 9 ((¬ 𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = ((0g𝑅)(.r𝑅)(𝑌𝑗)))
623, 4, 14ringlz 18633 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ (𝑌𝑗) ∈ 𝐵) → ((0g𝑅)(.r𝑅)(𝑌𝑗)) = (0g𝑅))
6337, 47, 62syl2anc 694 . . . . . . . . . 10 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → ((0g𝑅)(.r𝑅)(𝑌𝑗)) = (0g𝑅))
6463adantl 481 . . . . . . . . 9 ((¬ 𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → ((0g𝑅)(.r𝑅)(𝑌𝑗)) = (0g𝑅))
65 eqcom 2658 . . . . . . . . . . . 12 (𝑖 = 𝑗𝑗 = 𝑖)
66 iffalse 4128 . . . . . . . . . . . 12 𝑗 = 𝑖 → if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)) = (0g𝑅))
6765, 66sylnbi 319 . . . . . . . . . . 11 𝑖 = 𝑗 → if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)) = (0g𝑅))
6867eqcomd 2657 . . . . . . . . . 10 𝑖 = 𝑗 → (0g𝑅) = if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
6968adantr 480 . . . . . . . . 9 ((¬ 𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (0g𝑅) = if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
7061, 64, 693eqtrd 2689 . . . . . . . 8 ((¬ 𝑖 = 𝑗 ∧ ((𝜑𝑖𝑁) ∧ 𝑗𝑁)) → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
7158, 70pm2.61ian 848 . . . . . . 7 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → (if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))(.r𝑅)(𝑌𝑗)) = if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
7232, 71eqtrd 2685 . . . . . 6 (((𝜑𝑖𝑁) ∧ 𝑗𝑁) → ((𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗)(.r𝑅)(𝑌𝑗)) = if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
7372mpteq2dva 4777 . . . . 5 ((𝜑𝑖𝑁) → (𝑗𝑁 ↦ ((𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗)(.r𝑅)(𝑌𝑗))) = (𝑗𝑁 ↦ if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅))))
7473oveq2d 6706 . . . 4 ((𝜑𝑖𝑁) → (𝑅 Σg (𝑗𝑁 ↦ ((𝑖(𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, (1r𝑅), (0g𝑅)))𝑗)(.r𝑅)(𝑌𝑗)))) = (𝑅 Σg (𝑗𝑁 ↦ if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))))
75 ringmnd 18602 . . . . . . 7 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
765, 75syl 17 . . . . . 6 (𝜑𝑅 ∈ Mnd)
7776adantr 480 . . . . 5 ((𝜑𝑖𝑁) → 𝑅 ∈ Mnd)
786adantr 480 . . . . 5 ((𝜑𝑖𝑁) → 𝑁 ∈ Fin)
79 eqid 2651 . . . . 5 (𝑗𝑁 ↦ if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅))) = (𝑗𝑁 ↦ if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))
80 ffvelrn 6397 . . . . . . . . . 10 ((𝑌:𝑁𝐵𝑖𝑁) → (𝑌𝑖) ∈ 𝐵)
8180, 3syl6eleq 2740 . . . . . . . . 9 ((𝑌:𝑁𝐵𝑖𝑁) → (𝑌𝑖) ∈ (Base‘𝑅))
8281ex 449 . . . . . . . 8 (𝑌:𝑁𝐵 → (𝑖𝑁 → (𝑌𝑖) ∈ (Base‘𝑅)))
8341, 82syl6bi 243 . . . . . . 7 (𝜑 → (𝑌 ∈ (𝐵𝑚 𝑁) → (𝑖𝑁 → (𝑌𝑖) ∈ (Base‘𝑅))))
8411, 83mpd 15 . . . . . 6 (𝜑 → (𝑖𝑁 → (𝑌𝑖) ∈ (Base‘𝑅)))
8584imp 444 . . . . 5 ((𝜑𝑖𝑁) → (𝑌𝑖) ∈ (Base‘𝑅))
8614, 77, 78, 25, 79, 85gsummptif1n0 18411 . . . 4 ((𝜑𝑖𝑁) → (𝑅 Σg (𝑗𝑁 ↦ if(𝑗 = 𝑖, (𝑌𝑖), (0g𝑅)))) = (𝑌𝑖))
8720, 74, 863eqtrd 2689 . . 3 ((𝜑𝑖𝑁) → (𝑅 Σg (𝑗𝑁 ↦ ((𝑖(1r𝐴)𝑗)(.r𝑅)(𝑌𝑗)))) = (𝑌𝑖))
8887mpteq2dva 4777 . 2 (𝜑 → (𝑖𝑁 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖(1r𝐴)𝑗)(.r𝑅)(𝑌𝑗))))) = (𝑖𝑁 ↦ (𝑌𝑖)))
89 ffn 6083 . . . . 5 (𝑌:𝑁𝐵𝑌 Fn 𝑁)
9041, 89syl6bi 243 . . . 4 (𝜑 → (𝑌 ∈ (𝐵𝑚 𝑁) → 𝑌 Fn 𝑁))
9111, 90mpd 15 . . 3 (𝜑𝑌 Fn 𝑁)
92 eqcom 2658 . . . 4 ((𝑖𝑁 ↦ (𝑌𝑖)) = 𝑌𝑌 = (𝑖𝑁 ↦ (𝑌𝑖)))
93 dffn5 6280 . . . 4 (𝑌 Fn 𝑁𝑌 = (𝑖𝑁 ↦ (𝑌𝑖)))
9492, 93bitr4i 267 . . 3 ((𝑖𝑁 ↦ (𝑌𝑖)) = 𝑌𝑌 Fn 𝑁)
9591, 94sylibr 224 . 2 (𝜑 → (𝑖𝑁 ↦ (𝑌𝑖)) = 𝑌)
9612, 88, 953eqtrd 2689 1 (𝜑 → ((1r𝐴) · 𝑌) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  ifcif 4119  cop 4216  cmpt 4762   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  𝑚 cmap 7899  Fincfn 7997  Basecbs 15904  .rcmulr 15989  0gc0g 16147   Σg cgsu 16148  Mndcmnd 17341  1rcur 18547  Ringcrg 18593   Mat cmat 20261   maVecMul cmvmul 20394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-ot 4219  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-hom 16013  df-cco 16014  df-0g 16149  df-gsum 16150  df-prds 16155  df-pws 16157  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-ghm 17705  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-subrg 18826  df-lmod 18913  df-lss 18981  df-sra 19220  df-rgmod 19221  df-dsmm 20124  df-frlm 20139  df-mamu 20238  df-mat 20262  df-mvmul 20395
This theorem is referenced by:  slesolinv  20534  slesolinvbi  20535
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