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Theorem lmodcom 20750
Description: Left module vector sum is commutative. (Contributed by GΓ©rard Lang, 25-Jun-2014.)
Hypotheses
Ref Expression
lmodcom.v 𝑉 = (Baseβ€˜π‘Š)
lmodcom.a + = (+gβ€˜π‘Š)
Assertion
Ref Expression
lmodcom ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (𝑋 + π‘Œ) = (π‘Œ + 𝑋))

Proof of Theorem lmodcom
StepHypRef Expression
1 simp1 1135 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ π‘Š ∈ LMod)
2 eqid 2731 . . . . . . . . . . 11 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
3 eqid 2731 . . . . . . . . . . 11 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
4 eqid 2731 . . . . . . . . . . 11 (1rβ€˜(Scalarβ€˜π‘Š)) = (1rβ€˜(Scalarβ€˜π‘Š))
52, 3, 4lmod1cl 20731 . . . . . . . . . 10 (π‘Š ∈ LMod β†’ (1rβ€˜(Scalarβ€˜π‘Š)) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
61, 5syl 17 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (1rβ€˜(Scalarβ€˜π‘Š)) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
7 eqid 2731 . . . . . . . . . 10 (+gβ€˜(Scalarβ€˜π‘Š)) = (+gβ€˜(Scalarβ€˜π‘Š))
82, 3, 7lmodacl 20714 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ (1rβ€˜(Scalarβ€˜π‘Š)) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ (1rβ€˜(Scalarβ€˜π‘Š)) ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ ((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š))) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
91, 6, 6, 8syl3anc 1370 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š))) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
10 simp2 1136 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ 𝑋 ∈ 𝑉)
11 simp3 1137 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ π‘Œ ∈ 𝑉)
12 lmodcom.v . . . . . . . . 9 𝑉 = (Baseβ€˜π‘Š)
13 lmodcom.a . . . . . . . . 9 + = (+gβ€˜π‘Š)
14 eqid 2731 . . . . . . . . 9 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
1512, 13, 2, 14, 3lmodvsdi 20727 . . . . . . . 8 ((π‘Š ∈ LMod ∧ (((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š))) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š)))( ·𝑠 β€˜π‘Š)(𝑋 + π‘Œ)) = ((((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š)))( ·𝑠 β€˜π‘Š)𝑋) + (((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š)))( ·𝑠 β€˜π‘Š)π‘Œ)))
161, 9, 10, 11, 15syl13anc 1371 . . . . . . 7 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š)))( ·𝑠 β€˜π‘Š)(𝑋 + π‘Œ)) = ((((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š)))( ·𝑠 β€˜π‘Š)𝑋) + (((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š)))( ·𝑠 β€˜π‘Š)π‘Œ)))
1712, 13lmodvacl 20717 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (𝑋 + π‘Œ) ∈ 𝑉)
1812, 13, 2, 14, 3, 7lmodvsdir 20728 . . . . . . . 8 ((π‘Š ∈ LMod ∧ ((1rβ€˜(Scalarβ€˜π‘Š)) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ (1rβ€˜(Scalarβ€˜π‘Š)) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ (𝑋 + π‘Œ) ∈ 𝑉)) β†’ (((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š)))( ·𝑠 β€˜π‘Š)(𝑋 + π‘Œ)) = (((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)(𝑋 + π‘Œ)) + ((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)(𝑋 + π‘Œ))))
191, 6, 6, 17, 18syl13anc 1371 . . . . . . 7 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š)))( ·𝑠 β€˜π‘Š)(𝑋 + π‘Œ)) = (((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)(𝑋 + π‘Œ)) + ((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)(𝑋 + π‘Œ))))
2016, 19eqtr3d 2773 . . . . . 6 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š)))( ·𝑠 β€˜π‘Š)𝑋) + (((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š)))( ·𝑠 β€˜π‘Š)π‘Œ)) = (((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)(𝑋 + π‘Œ)) + ((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)(𝑋 + π‘Œ))))
2112, 13, 2, 14, 3, 7lmodvsdir 20728 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ ((1rβ€˜(Scalarβ€˜π‘Š)) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ (1rβ€˜(Scalarβ€˜π‘Š)) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑋 ∈ 𝑉)) β†’ (((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š)))( ·𝑠 β€˜π‘Š)𝑋) = (((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)𝑋) + ((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)𝑋)))
221, 6, 6, 10, 21syl13anc 1371 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š)))( ·𝑠 β€˜π‘Š)𝑋) = (((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)𝑋) + ((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)𝑋)))
2312, 2, 14, 4lmodvs1 20732 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ ((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)𝑋) = 𝑋)
241, 10, 23syl2anc 583 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)𝑋) = 𝑋)
2524, 24oveq12d 7430 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)𝑋) + ((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)𝑋)) = (𝑋 + 𝑋))
2622, 25eqtrd 2771 . . . . . . 7 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š)))( ·𝑠 β€˜π‘Š)𝑋) = (𝑋 + 𝑋))
2712, 13, 2, 14, 3, 7lmodvsdir 20728 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ ((1rβ€˜(Scalarβ€˜π‘Š)) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ (1rβ€˜(Scalarβ€˜π‘Š)) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘Œ ∈ 𝑉)) β†’ (((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š)))( ·𝑠 β€˜π‘Š)π‘Œ) = (((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)π‘Œ) + ((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)π‘Œ)))
281, 6, 6, 11, 27syl13anc 1371 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š)))( ·𝑠 β€˜π‘Š)π‘Œ) = (((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)π‘Œ) + ((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)π‘Œ)))
2912, 2, 14, 4lmodvs1 20732 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ ((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)π‘Œ) = π‘Œ)
301, 11, 29syl2anc 583 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)π‘Œ) = π‘Œ)
3130, 30oveq12d 7430 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)π‘Œ) + ((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)π‘Œ)) = (π‘Œ + π‘Œ))
3228, 31eqtrd 2771 . . . . . . 7 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š)))( ·𝑠 β€˜π‘Š)π‘Œ) = (π‘Œ + π‘Œ))
3326, 32oveq12d 7430 . . . . . 6 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š)))( ·𝑠 β€˜π‘Š)𝑋) + (((1rβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(1rβ€˜(Scalarβ€˜π‘Š)))( ·𝑠 β€˜π‘Š)π‘Œ)) = ((𝑋 + 𝑋) + (π‘Œ + π‘Œ)))
3412, 2, 14, 4lmodvs1 20732 . . . . . . . 8 ((π‘Š ∈ LMod ∧ (𝑋 + π‘Œ) ∈ 𝑉) β†’ ((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)(𝑋 + π‘Œ)) = (𝑋 + π‘Œ))
351, 17, 34syl2anc 583 . . . . . . 7 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)(𝑋 + π‘Œ)) = (𝑋 + π‘Œ))
3635, 35oveq12d 7430 . . . . . 6 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)(𝑋 + π‘Œ)) + ((1rβ€˜(Scalarβ€˜π‘Š))( ·𝑠 β€˜π‘Š)(𝑋 + π‘Œ))) = ((𝑋 + π‘Œ) + (𝑋 + π‘Œ)))
3720, 33, 363eqtr3d 2779 . . . . 5 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((𝑋 + 𝑋) + (π‘Œ + π‘Œ)) = ((𝑋 + π‘Œ) + (𝑋 + π‘Œ)))
3812, 13lmodvacl 20717 . . . . . . 7 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) β†’ (𝑋 + 𝑋) ∈ 𝑉)
391, 10, 10, 38syl3anc 1370 . . . . . 6 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (𝑋 + 𝑋) ∈ 𝑉)
4012, 13lmodass 20718 . . . . . 6 ((π‘Š ∈ LMod ∧ ((𝑋 + 𝑋) ∈ 𝑉 ∧ π‘Œ ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (((𝑋 + 𝑋) + π‘Œ) + π‘Œ) = ((𝑋 + 𝑋) + (π‘Œ + π‘Œ)))
411, 39, 11, 11, 40syl13anc 1371 . . . . 5 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (((𝑋 + 𝑋) + π‘Œ) + π‘Œ) = ((𝑋 + 𝑋) + (π‘Œ + π‘Œ)))
4212, 13lmodass 20718 . . . . . 6 ((π‘Š ∈ LMod ∧ ((𝑋 + π‘Œ) ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (((𝑋 + π‘Œ) + 𝑋) + π‘Œ) = ((𝑋 + π‘Œ) + (𝑋 + π‘Œ)))
431, 17, 10, 11, 42syl13anc 1371 . . . . 5 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (((𝑋 + π‘Œ) + 𝑋) + π‘Œ) = ((𝑋 + π‘Œ) + (𝑋 + π‘Œ)))
4437, 41, 433eqtr4d 2781 . . . 4 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (((𝑋 + 𝑋) + π‘Œ) + π‘Œ) = (((𝑋 + π‘Œ) + 𝑋) + π‘Œ))
45 lmodgrp 20709 . . . . . 6 (π‘Š ∈ LMod β†’ π‘Š ∈ Grp)
461, 45syl 17 . . . . 5 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ π‘Š ∈ Grp)
4712, 13lmodvacl 20717 . . . . . 6 ((π‘Š ∈ LMod ∧ (𝑋 + 𝑋) ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((𝑋 + 𝑋) + π‘Œ) ∈ 𝑉)
481, 39, 11, 47syl3anc 1370 . . . . 5 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((𝑋 + 𝑋) + π‘Œ) ∈ 𝑉)
4912, 13lmodvacl 20717 . . . . . 6 ((π‘Š ∈ LMod ∧ (𝑋 + π‘Œ) ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) β†’ ((𝑋 + π‘Œ) + 𝑋) ∈ 𝑉)
501, 17, 10, 49syl3anc 1370 . . . . 5 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((𝑋 + π‘Œ) + 𝑋) ∈ 𝑉)
5112, 13grprcan 18901 . . . . 5 ((π‘Š ∈ Grp ∧ (((𝑋 + 𝑋) + π‘Œ) ∈ 𝑉 ∧ ((𝑋 + π‘Œ) + 𝑋) ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ ((((𝑋 + 𝑋) + π‘Œ) + π‘Œ) = (((𝑋 + π‘Œ) + 𝑋) + π‘Œ) ↔ ((𝑋 + 𝑋) + π‘Œ) = ((𝑋 + π‘Œ) + 𝑋)))
5246, 48, 50, 11, 51syl13anc 1371 . . . 4 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((((𝑋 + 𝑋) + π‘Œ) + π‘Œ) = (((𝑋 + π‘Œ) + 𝑋) + π‘Œ) ↔ ((𝑋 + 𝑋) + π‘Œ) = ((𝑋 + π‘Œ) + 𝑋)))
5344, 52mpbid 231 . . 3 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((𝑋 + 𝑋) + π‘Œ) = ((𝑋 + π‘Œ) + 𝑋))
5412, 13lmodass 20718 . . . 4 ((π‘Š ∈ LMod ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ ((𝑋 + 𝑋) + π‘Œ) = (𝑋 + (𝑋 + π‘Œ)))
551, 10, 10, 11, 54syl13anc 1371 . . 3 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((𝑋 + 𝑋) + π‘Œ) = (𝑋 + (𝑋 + π‘Œ)))
5612, 13lmodass 20718 . . . 4 ((π‘Š ∈ LMod ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) β†’ ((𝑋 + π‘Œ) + 𝑋) = (𝑋 + (π‘Œ + 𝑋)))
571, 10, 11, 10, 56syl13anc 1371 . . 3 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((𝑋 + π‘Œ) + 𝑋) = (𝑋 + (π‘Œ + 𝑋)))
5853, 55, 573eqtr3d 2779 . 2 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (𝑋 + (𝑋 + π‘Œ)) = (𝑋 + (π‘Œ + 𝑋)))
5912, 13lmodvacl 20717 . . . 4 ((π‘Š ∈ LMod ∧ π‘Œ ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) β†’ (π‘Œ + 𝑋) ∈ 𝑉)
60593com23 1125 . . 3 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (π‘Œ + 𝑋) ∈ 𝑉)
6112, 13lmodlcan 20719 . . 3 ((π‘Š ∈ LMod ∧ ((𝑋 + π‘Œ) ∈ 𝑉 ∧ (π‘Œ + 𝑋) ∈ 𝑉 ∧ 𝑋 ∈ 𝑉)) β†’ ((𝑋 + (𝑋 + π‘Œ)) = (𝑋 + (π‘Œ + 𝑋)) ↔ (𝑋 + π‘Œ) = (π‘Œ + 𝑋)))
621, 17, 60, 10, 61syl13anc 1371 . 2 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ ((𝑋 + (𝑋 + π‘Œ)) = (𝑋 + (π‘Œ + 𝑋)) ↔ (𝑋 + π‘Œ) = (π‘Œ + 𝑋)))
6358, 62mpbid 231 1 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (𝑋 + π‘Œ) = (π‘Œ + 𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  β€˜cfv 6543  (class class class)co 7412  Basecbs 17151  +gcplusg 17204  Scalarcsca 17207   ·𝑠 cvsca 17208  Grpcgrp 18861  1rcur 20082  LModclmod 20702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-plusg 17217  df-0g 17394  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-grp 18864  df-minusg 18865  df-mgp 20036  df-ur 20083  df-ring 20136  df-lmod 20704
This theorem is referenced by:  lmodabl  20751  lssvsubcl  20787  lssvancl2  20789  lspsolv  20990  lflsub  38401  lcfrlem21  40898  lcfrlem42  40919  mapdindp4  41058
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