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Theorem mendring 37229
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 37221 . . 3 (𝑀 LMHom 𝑀) = (Base‘𝐴)
32a1i 11 . 2 (𝑀 ∈ LMod → (𝑀 LMHom 𝑀) = (Base‘𝐴))
4 eqidd 2627 . 2 (𝑀 ∈ LMod → (+g𝐴) = (+g𝐴))
5 eqidd 2627 . 2 (𝑀 ∈ LMod → (.r𝐴) = (.r𝐴))
6 eqid 2626 . . . . . 6 (+g𝑀) = (+g𝑀)
7 eqid 2626 . . . . . 6 (+g𝐴) = (+g𝐴)
81, 2, 6, 7mendplusg 37223 . . . . 5 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g𝐴)𝑦) = (𝑥𝑓 (+g𝑀)𝑦))
96lmhmplusg 18958 . . . . 5 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥𝑓 (+g𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
108, 9eqeltrd 2704 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
11103adant1 1077 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
12 simpr1 1065 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (𝑀 LMHom 𝑀))
13 simpr2 1066 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (𝑀 LMHom 𝑀))
1412, 13, 9syl2anc 692 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥𝑓 (+g𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
15 simpr3 1067 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ (𝑀 LMHom 𝑀))
161, 2, 6, 7mendplusg 37223 . . . . 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 692 . . . 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 692 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(+g𝐴)𝑦) = (𝑥𝑓 (+g𝑀)𝑦))
1918oveq1d 6620 . . . 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 18958 . . . . . . 7 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦𝑓 (+g𝑀)𝑧) ∈ (𝑀 LMHom 𝑀))
2113, 15, 20syl2anc 692 . . . . . 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 37223 . . . . . 6 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ (𝑦𝑓 (+g𝑀)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g𝐴)(𝑦𝑓 (+g𝑀)𝑧)) = (𝑥𝑓 (+g𝑀)(𝑦𝑓 (+g𝑀)𝑧)))
2312, 21, 22syl2anc 692 . . . . 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 37223 . . . . . . 7 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(+g𝐴)𝑧) = (𝑦𝑓 (+g𝑀)𝑧))
2513, 15, 24syl2anc 692 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(+g𝐴)𝑧) = (𝑦𝑓 (+g𝑀)𝑧))
2625oveq2d 6621 . . . . 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 18786 . . . . . . . 8 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
28 grpmnd 17345 . . . . . . . 8 (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
2927, 28syl 17 . . . . . . 7 (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
3029adantr 481 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑀 ∈ Mnd)
31 eqid 2626 . . . . . . . . 9 (Base‘𝑀) = (Base‘𝑀)
3231, 31lmhmf 18948 . . . . . . . 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 6160 . . . . . . . 8 (Base‘𝑀) ∈ V
3534, 34elmap 7831 . . . . . . 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 18948 . . . . . . . 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 7831 . . . . . . 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 18948 . . . . . . . 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 7831 . . . . . . 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 20112 . . . . . 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 1325 . . . . 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 2670 . . . 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 2670 . . 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 2627 . . . 4 (𝑀 ∈ LMod → (Scalar‘𝑀) = (Scalar‘𝑀))
51 eqid 2626 . . . . 5 (0g𝑀) = (0g𝑀)
52 eqid 2626 . . . . 5 (Scalar‘𝑀) = (Scalar‘𝑀)
5351, 31, 52, 520lmhm 18954 . . . 4 ((𝑀 ∈ LMod ∧ 𝑀 ∈ LMod ∧ (Scalar‘𝑀) = (Scalar‘𝑀)) → ((Base‘𝑀) × {(0g𝑀)}) ∈ (𝑀 LMHom 𝑀))
5449, 49, 50, 53syl3anc 1323 . . 3 (𝑀 ∈ LMod → ((Base‘𝑀) × {(0g𝑀)}) ∈ (𝑀 LMHom 𝑀))
551, 2, 6, 7mendplusg 37223 . . . . 5 ((((Base‘𝑀) × {(0g𝑀)}) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g𝑀)})(+g𝐴)𝑥) = (((Base‘𝑀) × {(0g𝑀)}) ∘𝑓 (+g𝑀)𝑥))
5654, 55sylan 488 . . . 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 20113 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑥 ∈ ((Base‘𝑀) ↑𝑚 (Base‘𝑀))) → (((Base‘𝑀) × {(0g𝑀)}) ∘𝑓 (+g𝑀)𝑥) = 𝑥)
5929, 57, 58syl2an 494 . . . 4 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g𝑀)}) ∘𝑓 (+g𝑀)𝑥) = 𝑥)
6056, 59eqtrd 2660 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g𝑀)})(+g𝐴)𝑥) = 𝑥)
61 eqid 2626 . . . . 5 (invg𝑀) = (invg𝑀)
6261invlmhm 18956 . . . 4 (𝑀 ∈ LMod → (invg𝑀) ∈ (𝑀 LMHom 𝑀))
63 lmhmco 18957 . . . 4 (((invg𝑀) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((invg𝑀) ∘ 𝑥) ∈ (𝑀 LMHom 𝑀))
6462, 63sylan 488 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((invg𝑀) ∘ 𝑥) ∈ (𝑀 LMHom 𝑀))
651, 2, 6, 7mendplusg 37223 . . . . 5 ((((invg𝑀) ∘ 𝑥) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg𝑀) ∘ 𝑥)(+g𝐴)𝑥) = (((invg𝑀) ∘ 𝑥) ∘𝑓 (+g𝑀)𝑥))
6664, 65sylancom 700 . . . 4 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg𝑀) ∘ 𝑥)(+g𝐴)𝑥) = (((invg𝑀) ∘ 𝑥) ∘𝑓 (+g𝑀)𝑥))
6731, 6, 61, 51grpvlinv 20115 . . . . 5 ((𝑀 ∈ Grp ∧ 𝑥 ∈ ((Base‘𝑀) ↑𝑚 (Base‘𝑀))) → (((invg𝑀) ∘ 𝑥) ∘𝑓 (+g𝑀)𝑥) = ((Base‘𝑀) × {(0g𝑀)}))
6827, 57, 67syl2an 494 . . . 4 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg𝑀) ∘ 𝑥) ∘𝑓 (+g𝑀)𝑥) = ((Base‘𝑀) × {(0g𝑀)}))
6966, 68eqtrd 2660 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg𝑀) ∘ 𝑥)(+g𝐴)𝑥) = ((Base‘𝑀) × {(0g𝑀)}))
703, 4, 11, 48, 54, 60, 64, 69isgrpd 17360 . 2 (𝑀 ∈ LMod → 𝐴 ∈ Grp)
71 eqid 2626 . . . . 5 (.r𝐴) = (.r𝐴)
721, 2, 71mendmulr 37225 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)𝑦) = (𝑥𝑦))
73 lmhmco 18957 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥𝑦) ∈ (𝑀 LMHom 𝑀))
7472, 73eqeltrd 2704 . . 3 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
75743adant1 1077 . 2 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
76 coass 5616 . . 3 ((𝑥𝑦) ∘ 𝑧) = (𝑥 ∘ (𝑦𝑧))
7712, 13, 72syl2anc 692 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)𝑦) = (𝑥𝑦))
7877oveq1d 6620 . . . 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 692 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥𝑦) ∈ (𝑀 LMHom 𝑀))
801, 2, 71mendmulr 37225 . . . . 5 (((𝑥𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥𝑦)(.r𝐴)𝑧) = ((𝑥𝑦) ∘ 𝑧))
8179, 15, 80syl2anc 692 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥𝑦)(.r𝐴)𝑧) = ((𝑥𝑦) ∘ 𝑧))
8278, 81eqtrd 2660 . . 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 37225 . . . . . 6 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(.r𝐴)𝑧) = (𝑦𝑧))
8413, 15, 83syl2anc 692 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r𝐴)𝑧) = (𝑦𝑧))
8584oveq2d 6621 . . . 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 18957 . . . . . 6 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦𝑧) ∈ (𝑀 LMHom 𝑀))
8713, 15, 86syl2anc 692 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦𝑧) ∈ (𝑀 LMHom 𝑀))
881, 2, 71mendmulr 37225 . . . . 5 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ (𝑦𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)(𝑦𝑧)) = (𝑥 ∘ (𝑦𝑧)))
8912, 87, 88syl2anc 692 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)(𝑦𝑧)) = (𝑥 ∘ (𝑦𝑧)))
9085, 89eqtrd 2660 . . 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 2686 . 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 37225 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ (𝑦𝑓 (+g𝑀)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)(𝑦𝑓 (+g𝑀)𝑧)) = (𝑥 ∘ (𝑦𝑓 (+g𝑀)𝑧)))
9312, 21, 92syl2anc 692 . . 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 6621 . . 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 18957 . . . . . 6 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥𝑧) ∈ (𝑀 LMHom 𝑀))
9612, 15, 95syl2anc 692 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥𝑧) ∈ (𝑀 LMHom 𝑀))
971, 2, 6, 7mendplusg 37223 . . . . 5 (((𝑥𝑦) ∈ (𝑀 LMHom 𝑀) ∧ (𝑥𝑧) ∈ (𝑀 LMHom 𝑀)) → ((𝑥𝑦)(+g𝐴)(𝑥𝑧)) = ((𝑥𝑦) ∘𝑓 (+g𝑀)(𝑥𝑧)))
9879, 96, 97syl2anc 692 . . . 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 37225 . . . . . 6 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)𝑧) = (𝑥𝑧))
10012, 15, 99syl2anc 692 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)𝑧) = (𝑥𝑧))
10177, 100oveq12d 6623 . . . 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 18945 . . . . . 6 (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥 ∈ (𝑀 GrpHom 𝑀))
103 ghmmhm 17586 . . . . . 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 20117 . . . . 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 1323 . . . 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 2670 . . 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 2670 . 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 37225 . . . 4 (((𝑥𝑓 (+g𝑀)𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥𝑓 (+g𝑀)𝑦)(.r𝐴)𝑧) = ((𝑥𝑓 (+g𝑀)𝑦) ∘ 𝑧))
11014, 15, 109syl2anc 692 . . 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 6620 . . 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 37223 . . . . 5 (((𝑥𝑧) ∈ (𝑀 LMHom 𝑀) ∧ (𝑦𝑧) ∈ (𝑀 LMHom 𝑀)) → ((𝑥𝑧)(+g𝐴)(𝑦𝑧)) = ((𝑥𝑧) ∘𝑓 (+g𝑀)(𝑦𝑧)))
11396, 87, 112syl2anc 692 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥𝑧)(+g𝐴)(𝑦𝑧)) = ((𝑥𝑧) ∘𝑓 (+g𝑀)(𝑦𝑧)))
114100, 84oveq12d 6623 . . . 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 6004 . . . . . 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 6004 . . . . . 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 3805 . . . . 5 ((Base‘𝑀) ∩ (Base‘𝑀)) = (Base‘𝑀)
121116, 118, 42, 119, 119, 119, 120ofco 6871 . . . 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 2670 . . 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 2670 . 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 18955 . 2 (𝑀 ∈ LMod → ( I ↾ (Base‘𝑀)) ∈ (𝑀 LMHom 𝑀))
1251, 2, 71mendmulr 37225 . . . 4 ((( I ↾ (Base‘𝑀)) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (( I ↾ (Base‘𝑀))(.r𝐴)𝑥) = (( I ↾ (Base‘𝑀)) ∘ 𝑥))
126124, 125sylan 488 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (( I ↾ (Base‘𝑀))(.r𝐴)𝑥) = (( I ↾ (Base‘𝑀)) ∘ 𝑥))
12732adantl 482 . . . 4 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
128 fcoi2 6038 . . . 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 2660 . 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 37225 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ ( I ↾ (Base‘𝑀)) ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)( I ↾ (Base‘𝑀))) = (𝑥 ∘ ( I ↾ (Base‘𝑀))))
133131, 124, 132syl2anr 495 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)( I ↾ (Base‘𝑀))) = (𝑥 ∘ ( I ↾ (Base‘𝑀))))
134 fcoi1 6037 . . . 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 2660 . 2 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)( I ↾ (Base‘𝑀))) = 𝑥)
1373, 4, 5, 70, 75, 91, 108, 123, 124, 130, 136isringd 18501 1 (𝑀 ∈ LMod → 𝐴 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  Vcvv 3191  {csn 4153   I cid 4989   × cxp 5077  cres 5081  ccom 5083   Fn wfn 5845  wf 5846  cfv 5850  (class class class)co 6605  𝑓 cof 6849  𝑚 cmap 7803  Basecbs 15776  +gcplusg 15857  .rcmulr 15858  Scalarcsca 15860  0gc0g 16016  Mndcmnd 17210   MndHom cmhm 17249  Grpcgrp 17338  invgcminusg 17339   GrpHom cghm 17573  Ringcrg 18463  LModclmod 18779   LMHom clmhm 18933  MEndocmend 37212
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-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-uni 4408  df-int 4446  df-iun 4492  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-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-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-of 6851  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-map 7805  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  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-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-struct 15778  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-plusg 15870  df-mulr 15871  df-sca 15873  df-vsca 15874  df-0g 16018  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-mhm 17251  df-grp 17341  df-minusg 17342  df-ghm 17574  df-cmn 18111  df-abl 18112  df-mgp 18406  df-ur 18418  df-ring 18465  df-lmod 18781  df-lmhm 18936  df-mend 37213
This theorem is referenced by:  mendlmod  37230  mendassa  37231
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