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Theorem mendring 38079
 Description: The module endomorphism algebra is a ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypothesis
Ref Expression
mendassa.a 𝐴 = (MEndo‘𝑀)
Assertion
Ref Expression
mendring (𝑀 ∈ LMod → 𝐴 ∈ Ring)

Proof of Theorem mendring
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mendassa.a . . . 4 𝐴 = (MEndo‘𝑀)
21mendbas 38071 . . 3 (𝑀 LMHom 𝑀) = (Base‘𝐴)
32a1i 11 . 2 (𝑀 ∈ LMod → (𝑀 LMHom 𝑀) = (Base‘𝐴))
4 eqidd 2652 . 2 (𝑀 ∈ LMod → (+g𝐴) = (+g𝐴))
5 eqidd 2652 . 2 (𝑀 ∈ LMod → (.r𝐴) = (.r𝐴))
6 eqid 2651 . . . . . 6 (+g𝑀) = (+g𝑀)
7 eqid 2651 . . . . . 6 (+g𝐴) = (+g𝐴)
81, 2, 6, 7mendplusg 38073 . . . . 5 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g𝐴)𝑦) = (𝑥𝑓 (+g𝑀)𝑦))
96lmhmplusg 19092 . . . . 5 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥𝑓 (+g𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
108, 9eqeltrd 2730 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
11103adant1 1099 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
12 simpr1 1087 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (𝑀 LMHom 𝑀))
13 simpr2 1088 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (𝑀 LMHom 𝑀))
1412, 13, 9syl2anc 694 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥𝑓 (+g𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
15 simpr3 1089 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ (𝑀 LMHom 𝑀))
161, 2, 6, 7mendplusg 38073 . . . . 5 (((𝑥𝑓 (+g𝑀)𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥𝑓 (+g𝑀)𝑦)(+g𝐴)𝑧) = ((𝑥𝑓 (+g𝑀)𝑦) ∘𝑓 (+g𝑀)𝑧))
1714, 15, 16syl2anc 694 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥𝑓 (+g𝑀)𝑦)(+g𝐴)𝑧) = ((𝑥𝑓 (+g𝑀)𝑦) ∘𝑓 (+g𝑀)𝑧))
1812, 13, 8syl2anc 694 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(+g𝐴)𝑦) = (𝑥𝑓 (+g𝑀)𝑦))
1918oveq1d 6705 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(+g𝐴)𝑦)(+g𝐴)𝑧) = ((𝑥𝑓 (+g𝑀)𝑦)(+g𝐴)𝑧))
206lmhmplusg 19092 . . . . . . 7 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦𝑓 (+g𝑀)𝑧) ∈ (𝑀 LMHom 𝑀))
2113, 15, 20syl2anc 694 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦𝑓 (+g𝑀)𝑧) ∈ (𝑀 LMHom 𝑀))
221, 2, 6, 7mendplusg 38073 . . . . . 6 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ (𝑦𝑓 (+g𝑀)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g𝐴)(𝑦𝑓 (+g𝑀)𝑧)) = (𝑥𝑓 (+g𝑀)(𝑦𝑓 (+g𝑀)𝑧)))
2312, 21, 22syl2anc 694 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(+g𝐴)(𝑦𝑓 (+g𝑀)𝑧)) = (𝑥𝑓 (+g𝑀)(𝑦𝑓 (+g𝑀)𝑧)))
241, 2, 6, 7mendplusg 38073 . . . . . . 7 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(+g𝐴)𝑧) = (𝑦𝑓 (+g𝑀)𝑧))
2513, 15, 24syl2anc 694 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(+g𝐴)𝑧) = (𝑦𝑓 (+g𝑀)𝑧))
2625oveq2d 6706 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(+g𝐴)(𝑦(+g𝐴)𝑧)) = (𝑥(+g𝐴)(𝑦𝑓 (+g𝑀)𝑧)))
27 lmodgrp 18918 . . . . . . . 8 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
28 grpmnd 17476 . . . . . . . 8 (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
2927, 28syl 17 . . . . . . 7 (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
3029adantr 480 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑀 ∈ Mnd)
31 eqid 2651 . . . . . . . . 9 (Base‘𝑀) = (Base‘𝑀)
3231, 31lmhmf 19082 . . . . . . . 8 (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
3312, 32syl 17 . . . . . . 7 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
34 fvex 6239 . . . . . . . 8 (Base‘𝑀) ∈ V
3534, 34elmap 7928 . . . . . . 7 (𝑥 ∈ ((Base‘𝑀) ↑𝑚 (Base‘𝑀)) ↔ 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
3633, 35sylibr 224 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ ((Base‘𝑀) ↑𝑚 (Base‘𝑀)))
3731, 31lmhmf 19082 . . . . . . . 8 (𝑦 ∈ (𝑀 LMHom 𝑀) → 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
3813, 37syl 17 . . . . . . 7 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
3934, 34elmap 7928 . . . . . . 7 (𝑦 ∈ ((Base‘𝑀) ↑𝑚 (Base‘𝑀)) ↔ 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
4038, 39sylibr 224 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ ((Base‘𝑀) ↑𝑚 (Base‘𝑀)))
4131, 31lmhmf 19082 . . . . . . . 8 (𝑧 ∈ (𝑀 LMHom 𝑀) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
4215, 41syl 17 . . . . . . 7 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
4334, 34elmap 7928 . . . . . . 7 (𝑧 ∈ ((Base‘𝑀) ↑𝑚 (Base‘𝑀)) ↔ 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
4442, 43sylibr 224 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ ((Base‘𝑀) ↑𝑚 (Base‘𝑀)))
4531, 6mndvass 20246 . . . . . 6 ((𝑀 ∈ Mnd ∧ (𝑥 ∈ ((Base‘𝑀) ↑𝑚 (Base‘𝑀)) ∧ 𝑦 ∈ ((Base‘𝑀) ↑𝑚 (Base‘𝑀)) ∧ 𝑧 ∈ ((Base‘𝑀) ↑𝑚 (Base‘𝑀)))) → ((𝑥𝑓 (+g𝑀)𝑦) ∘𝑓 (+g𝑀)𝑧) = (𝑥𝑓 (+g𝑀)(𝑦𝑓 (+g𝑀)𝑧)))
4630, 36, 40, 44, 45syl13anc 1368 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥𝑓 (+g𝑀)𝑦) ∘𝑓 (+g𝑀)𝑧) = (𝑥𝑓 (+g𝑀)(𝑦𝑓 (+g𝑀)𝑧)))
4723, 26, 463eqtr4d 2695 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(+g𝐴)(𝑦(+g𝐴)𝑧)) = ((𝑥𝑓 (+g𝑀)𝑦) ∘𝑓 (+g𝑀)𝑧))
4817, 19, 473eqtr4d 2695 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(+g𝐴)𝑦)(+g𝐴)𝑧) = (𝑥(+g𝐴)(𝑦(+g𝐴)𝑧)))
49 id 22 . . . 4 (𝑀 ∈ LMod → 𝑀 ∈ LMod)
50 eqidd 2652 . . . 4 (𝑀 ∈ LMod → (Scalar‘𝑀) = (Scalar‘𝑀))
51 eqid 2651 . . . . 5 (0g𝑀) = (0g𝑀)
52 eqid 2651 . . . . 5 (Scalar‘𝑀) = (Scalar‘𝑀)
5351, 31, 52, 520lmhm 19088 . . . 4 ((𝑀 ∈ LMod ∧ 𝑀 ∈ LMod ∧ (Scalar‘𝑀) = (Scalar‘𝑀)) → ((Base‘𝑀) × {(0g𝑀)}) ∈ (𝑀 LMHom 𝑀))
5449, 49, 50, 53syl3anc 1366 . . 3 (𝑀 ∈ LMod → ((Base‘𝑀) × {(0g𝑀)}) ∈ (𝑀 LMHom 𝑀))
551, 2, 6, 7mendplusg 38073 . . . . 5 ((((Base‘𝑀) × {(0g𝑀)}) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g𝑀)})(+g𝐴)𝑥) = (((Base‘𝑀) × {(0g𝑀)}) ∘𝑓 (+g𝑀)𝑥))
5654, 55sylan 487 . . . 4 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g𝑀)})(+g𝐴)𝑥) = (((Base‘𝑀) × {(0g𝑀)}) ∘𝑓 (+g𝑀)𝑥))
5732, 35sylibr 224 . . . . 5 (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥 ∈ ((Base‘𝑀) ↑𝑚 (Base‘𝑀)))
5831, 6, 51mndvlid 20247 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑥 ∈ ((Base‘𝑀) ↑𝑚 (Base‘𝑀))) → (((Base‘𝑀) × {(0g𝑀)}) ∘𝑓 (+g𝑀)𝑥) = 𝑥)
5929, 57, 58syl2an 493 . . . 4 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g𝑀)}) ∘𝑓 (+g𝑀)𝑥) = 𝑥)
6056, 59eqtrd 2685 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g𝑀)})(+g𝐴)𝑥) = 𝑥)
61 eqid 2651 . . . . 5 (invg𝑀) = (invg𝑀)
6261invlmhm 19090 . . . 4 (𝑀 ∈ LMod → (invg𝑀) ∈ (𝑀 LMHom 𝑀))
63 lmhmco 19091 . . . 4 (((invg𝑀) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((invg𝑀) ∘ 𝑥) ∈ (𝑀 LMHom 𝑀))
6462, 63sylan 487 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((invg𝑀) ∘ 𝑥) ∈ (𝑀 LMHom 𝑀))
651, 2, 6, 7mendplusg 38073 . . . . 5 ((((invg𝑀) ∘ 𝑥) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg𝑀) ∘ 𝑥)(+g𝐴)𝑥) = (((invg𝑀) ∘ 𝑥) ∘𝑓 (+g𝑀)𝑥))
6664, 65sylancom 702 . . . 4 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg𝑀) ∘ 𝑥)(+g𝐴)𝑥) = (((invg𝑀) ∘ 𝑥) ∘𝑓 (+g𝑀)𝑥))
6731, 6, 61, 51grpvlinv 20249 . . . . 5 ((𝑀 ∈ Grp ∧ 𝑥 ∈ ((Base‘𝑀) ↑𝑚 (Base‘𝑀))) → (((invg𝑀) ∘ 𝑥) ∘𝑓 (+g𝑀)𝑥) = ((Base‘𝑀) × {(0g𝑀)}))
6827, 57, 67syl2an 493 . . . 4 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg𝑀) ∘ 𝑥) ∘𝑓 (+g𝑀)𝑥) = ((Base‘𝑀) × {(0g𝑀)}))
6966, 68eqtrd 2685 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg𝑀) ∘ 𝑥)(+g𝐴)𝑥) = ((Base‘𝑀) × {(0g𝑀)}))
703, 4, 11, 48, 54, 60, 64, 69isgrpd 17491 . 2 (𝑀 ∈ LMod → 𝐴 ∈ Grp)
71 eqid 2651 . . . . 5 (.r𝐴) = (.r𝐴)
721, 2, 71mendmulr 38075 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)𝑦) = (𝑥𝑦))
73 lmhmco 19091 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥𝑦) ∈ (𝑀 LMHom 𝑀))
7472, 73eqeltrd 2730 . . 3 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
75743adant1 1099 . 2 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
76 coass 5692 . . 3 ((𝑥𝑦) ∘ 𝑧) = (𝑥 ∘ (𝑦𝑧))
7712, 13, 72syl2anc 694 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)𝑦) = (𝑥𝑦))
7877oveq1d 6705 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r𝐴)𝑦)(.r𝐴)𝑧) = ((𝑥𝑦)(.r𝐴)𝑧))
7912, 13, 73syl2anc 694 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥𝑦) ∈ (𝑀 LMHom 𝑀))
801, 2, 71mendmulr 38075 . . . . 5 (((𝑥𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥𝑦)(.r𝐴)𝑧) = ((𝑥𝑦) ∘ 𝑧))
8179, 15, 80syl2anc 694 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥𝑦)(.r𝐴)𝑧) = ((𝑥𝑦) ∘ 𝑧))
8278, 81eqtrd 2685 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r𝐴)𝑦)(.r𝐴)𝑧) = ((𝑥𝑦) ∘ 𝑧))
831, 2, 71mendmulr 38075 . . . . . 6 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(.r𝐴)𝑧) = (𝑦𝑧))
8413, 15, 83syl2anc 694 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r𝐴)𝑧) = (𝑦𝑧))
8584oveq2d 6706 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)(𝑦(.r𝐴)𝑧)) = (𝑥(.r𝐴)(𝑦𝑧)))
86 lmhmco 19091 . . . . . 6 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦𝑧) ∈ (𝑀 LMHom 𝑀))
8713, 15, 86syl2anc 694 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦𝑧) ∈ (𝑀 LMHom 𝑀))
881, 2, 71mendmulr 38075 . . . . 5 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ (𝑦𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)(𝑦𝑧)) = (𝑥 ∘ (𝑦𝑧)))
8912, 87, 88syl2anc 694 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)(𝑦𝑧)) = (𝑥 ∘ (𝑦𝑧)))
9085, 89eqtrd 2685 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)(𝑦(.r𝐴)𝑧)) = (𝑥 ∘ (𝑦𝑧)))
9176, 82, 903eqtr4a 2711 . 2 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r𝐴)𝑦)(.r𝐴)𝑧) = (𝑥(.r𝐴)(𝑦(.r𝐴)𝑧)))
921, 2, 71mendmulr 38075 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ (𝑦𝑓 (+g𝑀)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)(𝑦𝑓 (+g𝑀)𝑧)) = (𝑥 ∘ (𝑦𝑓 (+g𝑀)𝑧)))
9312, 21, 92syl2anc 694 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)(𝑦𝑓 (+g𝑀)𝑧)) = (𝑥 ∘ (𝑦𝑓 (+g𝑀)𝑧)))
9425oveq2d 6706 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)(𝑦(+g𝐴)𝑧)) = (𝑥(.r𝐴)(𝑦𝑓 (+g𝑀)𝑧)))
95 lmhmco 19091 . . . . . 6 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥𝑧) ∈ (𝑀 LMHom 𝑀))
9612, 15, 95syl2anc 694 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥𝑧) ∈ (𝑀 LMHom 𝑀))
971, 2, 6, 7mendplusg 38073 . . . . 5 (((𝑥𝑦) ∈ (𝑀 LMHom 𝑀) ∧ (𝑥𝑧) ∈ (𝑀 LMHom 𝑀)) → ((𝑥𝑦)(+g𝐴)(𝑥𝑧)) = ((𝑥𝑦) ∘𝑓 (+g𝑀)(𝑥𝑧)))
9879, 96, 97syl2anc 694 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥𝑦)(+g𝐴)(𝑥𝑧)) = ((𝑥𝑦) ∘𝑓 (+g𝑀)(𝑥𝑧)))
991, 2, 71mendmulr 38075 . . . . . 6 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)𝑧) = (𝑥𝑧))
10012, 15, 99syl2anc 694 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)𝑧) = (𝑥𝑧))
10177, 100oveq12d 6708 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r𝐴)𝑦)(+g𝐴)(𝑥(.r𝐴)𝑧)) = ((𝑥𝑦)(+g𝐴)(𝑥𝑧)))
102 lmghm 19079 . . . . . 6 (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥 ∈ (𝑀 GrpHom 𝑀))
103 ghmmhm 17717 . . . . . 6 (𝑥 ∈ (𝑀 GrpHom 𝑀) → 𝑥 ∈ (𝑀 MndHom 𝑀))
10412, 102, 1033syl 18 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (𝑀 MndHom 𝑀))
10531, 6, 6mhmvlin 20251 . . . . 5 ((𝑥 ∈ (𝑀 MndHom 𝑀) ∧ 𝑦 ∈ ((Base‘𝑀) ↑𝑚 (Base‘𝑀)) ∧ 𝑧 ∈ ((Base‘𝑀) ↑𝑚 (Base‘𝑀))) → (𝑥 ∘ (𝑦𝑓 (+g𝑀)𝑧)) = ((𝑥𝑦) ∘𝑓 (+g𝑀)(𝑥𝑧)))
106104, 40, 44, 105syl3anc 1366 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥 ∘ (𝑦𝑓 (+g𝑀)𝑧)) = ((𝑥𝑦) ∘𝑓 (+g𝑀)(𝑥𝑧)))
10798, 101, 1063eqtr4d 2695 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r𝐴)𝑦)(+g𝐴)(𝑥(.r𝐴)𝑧)) = (𝑥 ∘ (𝑦𝑓 (+g𝑀)𝑧)))
10893, 94, 1073eqtr4d 2695 . 2 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)(𝑦(+g𝐴)𝑧)) = ((𝑥(.r𝐴)𝑦)(+g𝐴)(𝑥(.r𝐴)𝑧)))
1091, 2, 71mendmulr 38075 . . . 4 (((𝑥𝑓 (+g𝑀)𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥𝑓 (+g𝑀)𝑦)(.r𝐴)𝑧) = ((𝑥𝑓 (+g𝑀)𝑦) ∘ 𝑧))
11014, 15, 109syl2anc 694 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥𝑓 (+g𝑀)𝑦)(.r𝐴)𝑧) = ((𝑥𝑓 (+g𝑀)𝑦) ∘ 𝑧))
11118oveq1d 6705 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(+g𝐴)𝑦)(.r𝐴)𝑧) = ((𝑥𝑓 (+g𝑀)𝑦)(.r𝐴)𝑧))
1121, 2, 6, 7mendplusg 38073 . . . . 5 (((𝑥𝑧) ∈ (𝑀 LMHom 𝑀) ∧ (𝑦𝑧) ∈ (𝑀 LMHom 𝑀)) → ((𝑥𝑧)(+g𝐴)(𝑦𝑧)) = ((𝑥𝑧) ∘𝑓 (+g𝑀)(𝑦𝑧)))
11396, 87, 112syl2anc 694 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥𝑧)(+g𝐴)(𝑦𝑧)) = ((𝑥𝑧) ∘𝑓 (+g𝑀)(𝑦𝑧)))
114100, 84oveq12d 6708 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r𝐴)𝑧)(+g𝐴)(𝑦(.r𝐴)𝑧)) = ((𝑥𝑧)(+g𝐴)(𝑦𝑧)))
115 ffn 6083 . . . . . 6 (𝑥:(Base‘𝑀)⟶(Base‘𝑀) → 𝑥 Fn (Base‘𝑀))
11612, 32, 1153syl 18 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 Fn (Base‘𝑀))
117 ffn 6083 . . . . . 6 (𝑦:(Base‘𝑀)⟶(Base‘𝑀) → 𝑦 Fn (Base‘𝑀))
11813, 37, 1173syl 18 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 Fn (Base‘𝑀))
11934a1i 11 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (Base‘𝑀) ∈ V)
120 inidm 3855 . . . . 5 ((Base‘𝑀) ∩ (Base‘𝑀)) = (Base‘𝑀)
121116, 118, 42, 119, 119, 119, 120ofco 6959 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥𝑓 (+g𝑀)𝑦) ∘ 𝑧) = ((𝑥𝑧) ∘𝑓 (+g𝑀)(𝑦𝑧)))
122113, 114, 1213eqtr4d 2695 . . 3 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r𝐴)𝑧)(+g𝐴)(𝑦(.r𝐴)𝑧)) = ((𝑥𝑓 (+g𝑀)𝑦) ∘ 𝑧))
123110, 111, 1223eqtr4d 2695 . 2 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(+g𝐴)𝑦)(.r𝐴)𝑧) = ((𝑥(.r𝐴)𝑧)(+g𝐴)(𝑦(.r𝐴)𝑧)))
12431idlmhm 19089 . 2 (𝑀 ∈ LMod → ( I ↾ (Base‘𝑀)) ∈ (𝑀 LMHom 𝑀))
1251, 2, 71mendmulr 38075 . . . 4 ((( I ↾ (Base‘𝑀)) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (( I ↾ (Base‘𝑀))(.r𝐴)𝑥) = (( I ↾ (Base‘𝑀)) ∘ 𝑥))
126124, 125sylan 487 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (( I ↾ (Base‘𝑀))(.r𝐴)𝑥) = (( I ↾ (Base‘𝑀)) ∘ 𝑥))
12732adantl 481 . . . 4 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
128 fcoi2 6117 . . . 4 (𝑥:(Base‘𝑀)⟶(Base‘𝑀) → (( I ↾ (Base‘𝑀)) ∘ 𝑥) = 𝑥)
129127, 128syl 17 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (( I ↾ (Base‘𝑀)) ∘ 𝑥) = 𝑥)
130126, 129eqtrd 2685 . 2 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (( I ↾ (Base‘𝑀))(.r𝐴)𝑥) = 𝑥)
131 id 22 . . . 4 (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥 ∈ (𝑀 LMHom 𝑀))
1321, 2, 71mendmulr 38075 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ ( I ↾ (Base‘𝑀)) ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)( I ↾ (Base‘𝑀))) = (𝑥 ∘ ( I ↾ (Base‘𝑀))))
133131, 124, 132syl2anr 494 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)( I ↾ (Base‘𝑀))) = (𝑥 ∘ ( I ↾ (Base‘𝑀))))
134 fcoi1 6116 . . . 4 (𝑥:(Base‘𝑀)⟶(Base‘𝑀) → (𝑥 ∘ ( I ↾ (Base‘𝑀))) = 𝑥)
135127, 134syl 17 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (𝑥 ∘ ( I ↾ (Base‘𝑀))) = 𝑥)
136133, 135eqtrd 2685 . 2 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)( I ↾ (Base‘𝑀))) = 𝑥)
1373, 4, 5, 70, 75, 91, 108, 123, 124, 130, 136isringd 18631 1 (𝑀 ∈ LMod → 𝐴 ∈ Ring)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  Vcvv 3231  {csn 4210   I cid 5052   × cxp 5141   ↾ cres 5145   ∘ ccom 5147   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ∘𝑓 cof 6937   ↑𝑚 cmap 7899  Basecbs 15904  +gcplusg 15988  .rcmulr 15989  Scalarcsca 15991  0gc0g 16147  Mndcmnd 17341   MndHom cmhm 17380  Grpcgrp 17469  invgcminusg 17470   GrpHom cghm 17704  Ringcrg 18593  LModclmod 18911   LMHom clmhm 19067  MEndocmend 38062 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-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-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-uni 4469  df-int 4508  df-iun 4554  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-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-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  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-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-grp 17472  df-minusg 17473  df-ghm 17705  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-lmod 18913  df-lmhm 19070  df-mend 38063 This theorem is referenced by:  mendlmod  38080  mendassa  38081
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