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Theorem mendring 38281
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 38273 . . 3 (𝑀 LMHom 𝑀) = (Base‘𝐴)
32a1i 11 . 2 (𝑀 ∈ LMod → (𝑀 LMHom 𝑀) = (Base‘𝐴))
4 eqidd 2771 . 2 (𝑀 ∈ LMod → (+g𝐴) = (+g𝐴))
5 eqidd 2771 . 2 (𝑀 ∈ LMod → (.r𝐴) = (.r𝐴))
6 eqid 2770 . . . . . 6 (+g𝑀) = (+g𝑀)
7 eqid 2770 . . . . . 6 (+g𝐴) = (+g𝐴)
81, 2, 6, 7mendplusg 38275 . . . . 5 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g𝐴)𝑦) = (𝑥𝑓 (+g𝑀)𝑦))
96lmhmplusg 19256 . . . . 5 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥𝑓 (+g𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
108, 9eqeltrd 2849 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
11103adant1 1123 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
12 simpr1 1232 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (𝑀 LMHom 𝑀))
13 simpr2 1234 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (𝑀 LMHom 𝑀))
1412, 13, 9syl2anc 565 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥𝑓 (+g𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
15 simpr3 1236 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ (𝑀 LMHom 𝑀))
161, 2, 6, 7mendplusg 38275 . . . . 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 565 . . . 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 565 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(+g𝐴)𝑦) = (𝑥𝑓 (+g𝑀)𝑦))
1918oveq1d 6807 . . . 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 19256 . . . . . . 7 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦𝑓 (+g𝑀)𝑧) ∈ (𝑀 LMHom 𝑀))
2113, 15, 20syl2anc 565 . . . . . 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 38275 . . . . . 6 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ (𝑦𝑓 (+g𝑀)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥(+g𝐴)(𝑦𝑓 (+g𝑀)𝑧)) = (𝑥𝑓 (+g𝑀)(𝑦𝑓 (+g𝑀)𝑧)))
2312, 21, 22syl2anc 565 . . . . 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 38275 . . . . . . 7 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(+g𝐴)𝑧) = (𝑦𝑓 (+g𝑀)𝑧))
2513, 15, 24syl2anc 565 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(+g𝐴)𝑧) = (𝑦𝑓 (+g𝑀)𝑧))
2625oveq2d 6808 . . . . 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 19079 . . . . . . . 8 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
28 grpmnd 17636 . . . . . . . 8 (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
2927, 28syl 17 . . . . . . 7 (𝑀 ∈ LMod → 𝑀 ∈ Mnd)
3029adantr 466 . . . . . 6 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑀 ∈ Mnd)
31 eqid 2770 . . . . . . . . 9 (Base‘𝑀) = (Base‘𝑀)
3231, 31lmhmf 19246 . . . . . . . 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 6342 . . . . . . . 8 (Base‘𝑀) ∈ V
3534, 34elmap 8037 . . . . . . 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 19246 . . . . . . . 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 8037 . . . . . . 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 19246 . . . . . . . 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 8037 . . . . . . 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 20414 . . . . . 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 1477 . . . . 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 2814 . . . 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 2814 . . 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 2771 . . . 4 (𝑀 ∈ LMod → (Scalar‘𝑀) = (Scalar‘𝑀))
51 eqid 2770 . . . . 5 (0g𝑀) = (0g𝑀)
52 eqid 2770 . . . . 5 (Scalar‘𝑀) = (Scalar‘𝑀)
5351, 31, 52, 520lmhm 19252 . . . 4 ((𝑀 ∈ LMod ∧ 𝑀 ∈ LMod ∧ (Scalar‘𝑀) = (Scalar‘𝑀)) → ((Base‘𝑀) × {(0g𝑀)}) ∈ (𝑀 LMHom 𝑀))
5449, 49, 50, 53syl3anc 1475 . . 3 (𝑀 ∈ LMod → ((Base‘𝑀) × {(0g𝑀)}) ∈ (𝑀 LMHom 𝑀))
551, 2, 6, 7mendplusg 38275 . . . . 5 ((((Base‘𝑀) × {(0g𝑀)}) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g𝑀)})(+g𝐴)𝑥) = (((Base‘𝑀) × {(0g𝑀)}) ∘𝑓 (+g𝑀)𝑥))
5654, 55sylan 561 . . . 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 20415 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑥 ∈ ((Base‘𝑀) ↑𝑚 (Base‘𝑀))) → (((Base‘𝑀) × {(0g𝑀)}) ∘𝑓 (+g𝑀)𝑥) = 𝑥)
5929, 57, 58syl2an 575 . . . 4 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g𝑀)}) ∘𝑓 (+g𝑀)𝑥) = 𝑥)
6056, 59eqtrd 2804 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(0g𝑀)})(+g𝐴)𝑥) = 𝑥)
61 eqid 2770 . . . . 5 (invg𝑀) = (invg𝑀)
6261invlmhm 19254 . . . 4 (𝑀 ∈ LMod → (invg𝑀) ∈ (𝑀 LMHom 𝑀))
63 lmhmco 19255 . . . 4 (((invg𝑀) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((invg𝑀) ∘ 𝑥) ∈ (𝑀 LMHom 𝑀))
6462, 63sylan 561 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((invg𝑀) ∘ 𝑥) ∈ (𝑀 LMHom 𝑀))
651, 2, 6, 7mendplusg 38275 . . . . 5 ((((invg𝑀) ∘ 𝑥) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg𝑀) ∘ 𝑥)(+g𝐴)𝑥) = (((invg𝑀) ∘ 𝑥) ∘𝑓 (+g𝑀)𝑥))
6664, 65sylancom 568 . . . 4 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg𝑀) ∘ 𝑥)(+g𝐴)𝑥) = (((invg𝑀) ∘ 𝑥) ∘𝑓 (+g𝑀)𝑥))
6731, 6, 61, 51grpvlinv 20417 . . . . 5 ((𝑀 ∈ Grp ∧ 𝑥 ∈ ((Base‘𝑀) ↑𝑚 (Base‘𝑀))) → (((invg𝑀) ∘ 𝑥) ∘𝑓 (+g𝑀)𝑥) = ((Base‘𝑀) × {(0g𝑀)}))
6827, 57, 67syl2an 575 . . . 4 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg𝑀) ∘ 𝑥) ∘𝑓 (+g𝑀)𝑥) = ((Base‘𝑀) × {(0g𝑀)}))
6966, 68eqtrd 2804 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((invg𝑀) ∘ 𝑥)(+g𝐴)𝑥) = ((Base‘𝑀) × {(0g𝑀)}))
703, 4, 11, 48, 54, 60, 64, 69isgrpd 17651 . 2 (𝑀 ∈ LMod → 𝐴 ∈ Grp)
71 eqid 2770 . . . . 5 (.r𝐴) = (.r𝐴)
721, 2, 71mendmulr 38277 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)𝑦) = (𝑥𝑦))
73 lmhmco 19255 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥𝑦) ∈ (𝑀 LMHom 𝑀))
7472, 73eqeltrd 2849 . . 3 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
75743adant1 1123 . 2 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
76 coass 5798 . . 3 ((𝑥𝑦) ∘ 𝑧) = (𝑥 ∘ (𝑦𝑧))
7712, 13, 72syl2anc 565 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)𝑦) = (𝑥𝑦))
7877oveq1d 6807 . . . 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 565 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥𝑦) ∈ (𝑀 LMHom 𝑀))
801, 2, 71mendmulr 38277 . . . . 5 (((𝑥𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥𝑦)(.r𝐴)𝑧) = ((𝑥𝑦) ∘ 𝑧))
8179, 15, 80syl2anc 565 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥𝑦)(.r𝐴)𝑧) = ((𝑥𝑦) ∘ 𝑧))
8278, 81eqtrd 2804 . . 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 38277 . . . . . 6 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(.r𝐴)𝑧) = (𝑦𝑧))
8413, 15, 83syl2anc 565 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(.r𝐴)𝑧) = (𝑦𝑧))
8584oveq2d 6808 . . . 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 19255 . . . . . 6 ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦𝑧) ∈ (𝑀 LMHom 𝑀))
8713, 15, 86syl2anc 565 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦𝑧) ∈ (𝑀 LMHom 𝑀))
881, 2, 71mendmulr 38277 . . . . 5 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ (𝑦𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)(𝑦𝑧)) = (𝑥 ∘ (𝑦𝑧)))
8912, 87, 88syl2anc 565 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)(𝑦𝑧)) = (𝑥 ∘ (𝑦𝑧)))
9085, 89eqtrd 2804 . . 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 2830 . 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 38277 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ (𝑦𝑓 (+g𝑀)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)(𝑦𝑓 (+g𝑀)𝑧)) = (𝑥 ∘ (𝑦𝑓 (+g𝑀)𝑧)))
9312, 21, 92syl2anc 565 . . 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 6808 . . 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 19255 . . . . . 6 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥𝑧) ∈ (𝑀 LMHom 𝑀))
9612, 15, 95syl2anc 565 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥𝑧) ∈ (𝑀 LMHom 𝑀))
971, 2, 6, 7mendplusg 38275 . . . . 5 (((𝑥𝑦) ∈ (𝑀 LMHom 𝑀) ∧ (𝑥𝑧) ∈ (𝑀 LMHom 𝑀)) → ((𝑥𝑦)(+g𝐴)(𝑥𝑧)) = ((𝑥𝑦) ∘𝑓 (+g𝑀)(𝑥𝑧)))
9879, 96, 97syl2anc 565 . . . 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 38277 . . . . . 6 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)𝑧) = (𝑥𝑧))
10012, 15, 99syl2anc 565 . . . . 5 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r𝐴)𝑧) = (𝑥𝑧))
10177, 100oveq12d 6810 . . . 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 19243 . . . . . 6 (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥 ∈ (𝑀 GrpHom 𝑀))
103 ghmmhm 17877 . . . . . 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 20419 . . . . 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 1475 . . . 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 2814 . . 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 2814 . 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 38277 . . . 4 (((𝑥𝑓 (+g𝑀)𝑦) ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥𝑓 (+g𝑀)𝑦)(.r𝐴)𝑧) = ((𝑥𝑓 (+g𝑀)𝑦) ∘ 𝑧))
11014, 15, 109syl2anc 565 . . 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 6807 . . 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 38275 . . . . 5 (((𝑥𝑧) ∈ (𝑀 LMHom 𝑀) ∧ (𝑦𝑧) ∈ (𝑀 LMHom 𝑀)) → ((𝑥𝑧)(+g𝐴)(𝑦𝑧)) = ((𝑥𝑧) ∘𝑓 (+g𝑀)(𝑦𝑧)))
11396, 87, 112syl2anc 565 . . . 4 ((𝑀 ∈ LMod ∧ (𝑥 ∈ (𝑀 LMHom 𝑀) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥𝑧)(+g𝐴)(𝑦𝑧)) = ((𝑥𝑧) ∘𝑓 (+g𝑀)(𝑦𝑧)))
114100, 84oveq12d 6810 . . . 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 6185 . . . . . 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 6185 . . . . . 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 3969 . . . . 5 ((Base‘𝑀) ∩ (Base‘𝑀)) = (Base‘𝑀)
121116, 118, 42, 119, 119, 119, 120ofco 7063 . . . 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 2814 . . 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 2814 . 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 19253 . 2 (𝑀 ∈ LMod → ( I ↾ (Base‘𝑀)) ∈ (𝑀 LMHom 𝑀))
1251, 2, 71mendmulr 38277 . . . 4 ((( I ↾ (Base‘𝑀)) ∈ (𝑀 LMHom 𝑀) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (( I ↾ (Base‘𝑀))(.r𝐴)𝑥) = (( I ↾ (Base‘𝑀)) ∘ 𝑥))
126124, 125sylan 561 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (( I ↾ (Base‘𝑀))(.r𝐴)𝑥) = (( I ↾ (Base‘𝑀)) ∘ 𝑥))
12732adantl 467 . . . 4 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
128 fcoi2 6219 . . . 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 2804 . 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 38277 . . . 4 ((𝑥 ∈ (𝑀 LMHom 𝑀) ∧ ( I ↾ (Base‘𝑀)) ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)( I ↾ (Base‘𝑀))) = (𝑥 ∘ ( I ↾ (Base‘𝑀))))
133131, 124, 132syl2anr 576 . . 3 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)( I ↾ (Base‘𝑀))) = (𝑥 ∘ ( I ↾ (Base‘𝑀))))
134 fcoi1 6218 . . . 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 2804 . 2 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (𝑥(.r𝐴)( I ↾ (Base‘𝑀))) = 𝑥)
1373, 4, 5, 70, 75, 91, 108, 123, 124, 130, 136isringd 18792 1 (𝑀 ∈ LMod → 𝐴 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1070   = wceq 1630  wcel 2144  Vcvv 3349  {csn 4314   I cid 5156   × cxp 5247  cres 5251  ccom 5253   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6792  𝑓 cof 7041  𝑚 cmap 8008  Basecbs 16063  +gcplusg 16148  .rcmulr 16149  Scalarcsca 16151  0gc0g 16307  Mndcmnd 17501   MndHom cmhm 17540  Grpcgrp 17629  invgcminusg 17630   GrpHom cghm 17864  Ringcrg 18754  LModclmod 19072   LMHom clmhm 19231  MEndocmend 38264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-of 7043  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-n0 11494  df-z 11579  df-uz 11888  df-fz 12533  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-plusg 16161  df-mulr 16162  df-sca 16164  df-vsca 16165  df-0g 16309  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-mhm 17542  df-grp 17632  df-minusg 17633  df-ghm 17865  df-cmn 18401  df-abl 18402  df-mgp 18697  df-ur 18709  df-ring 18756  df-lmod 19074  df-lmhm 19234  df-mend 38265
This theorem is referenced by:  mendlmod  38282  mendassa  38283
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