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Theorem lmod1 44854
Description: The (smallest) structure representing a zero module over an arbitrary ring. (Contributed by AV, 29-Apr-2019.)
Hypothesis
Ref Expression
lmod1.m 𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})
Assertion
Ref Expression
lmod1 ((𝐼𝑉𝑅 ∈ Ring) → 𝑀 ∈ LMod)
Distinct variable groups:   𝑥,𝐼,𝑦   𝑥,𝑅,𝑦   𝑥,𝑉,𝑦   𝑥,𝑀,𝑦

Proof of Theorem lmod1
Dummy variables 𝑟 𝑞 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2824 . . . . 5 {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}
21grp1 18208 . . . 4 (𝐼𝑉 → {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} ∈ Grp)
3 fvex 6676 . . . . . . 7 (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) ∈ V
4 lmod1.m . . . . . . . . 9 𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})
5 snex 5320 . . . . . . . . . . . . 13 {𝐼} ∈ V
61grpbase 16612 . . . . . . . . . . . . 13 ({𝐼} ∈ V → {𝐼} = (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}))
75, 6ax-mp 5 . . . . . . . . . . . 12 {𝐼} = (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})
87opeq2i 4793 . . . . . . . . . . 11 ⟨(Base‘ndx), {𝐼}⟩ = ⟨(Base‘ndx), (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩
9 tpeq1 4663 . . . . . . . . . . 11 (⟨(Base‘ndx), {𝐼}⟩ = ⟨(Base‘ndx), (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩ → {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} = {⟨(Base‘ndx), (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩})
108, 9ax-mp 5 . . . . . . . . . 10 {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} = {⟨(Base‘ndx), (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩}
1110uneq1i 4121 . . . . . . . . 9 ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩}) = ({⟨(Base‘ndx), (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})
124, 11eqtri 2847 . . . . . . . 8 𝑀 = ({⟨(Base‘ndx), (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})
1312lmodbase 16639 . . . . . . 7 ((Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) ∈ V → (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) = (Base‘𝑀))
143, 13ax-mp 5 . . . . . 6 (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) = (Base‘𝑀)
1514eqcomi 2833 . . . . 5 (Base‘𝑀) = (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})
16 fvex 6676 . . . . . . 7 (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) ∈ V
17 snex 5320 . . . . . . . . . . . . 13 {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} ∈ V
181grpplusg 16613 . . . . . . . . . . . . 13 ({⟨⟨𝐼, 𝐼⟩, 𝐼⟩} ∈ V → {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} = (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}))
1917, 18ax-mp 5 . . . . . . . . . . . 12 {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} = (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})
2019opeq2i 4793 . . . . . . . . . . 11 ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩ = ⟨(+g‘ndx), (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩
21 tpeq2 4664 . . . . . . . . . . 11 (⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩ = ⟨(+g‘ndx), (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩ → {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩, ⟨(Scalar‘ndx), 𝑅⟩})
2220, 21ax-mp 5 . . . . . . . . . 10 {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩, ⟨(Scalar‘ndx), 𝑅⟩}
2322uneq1i 4121 . . . . . . . . 9 ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩}) = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})
244, 23eqtri 2847 . . . . . . . 8 𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})
2524lmodplusg 16640 . . . . . . 7 ((+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) ∈ V → (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) = (+g𝑀))
2616, 25ax-mp 5 . . . . . 6 (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) = (+g𝑀)
2726eqcomi 2833 . . . . 5 (+g𝑀) = (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})
2815, 27grpprop 18121 . . . 4 (𝑀 ∈ Grp ↔ {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} ∈ Grp)
292, 28sylibr 237 . . 3 (𝐼𝑉𝑀 ∈ Grp)
3029adantr 484 . 2 ((𝐼𝑉𝑅 ∈ Ring) → 𝑀 ∈ Grp)
314lmodsca 16641 . . . . 5 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑀))
3231eqcomd 2830 . . . 4 (𝑅 ∈ Ring → (Scalar‘𝑀) = 𝑅)
3332adantl 485 . . 3 ((𝐼𝑉𝑅 ∈ Ring) → (Scalar‘𝑀) = 𝑅)
34 simpr 488 . . 3 ((𝐼𝑉𝑅 ∈ Ring) → 𝑅 ∈ Ring)
3533, 34eqeltrd 2916 . 2 ((𝐼𝑉𝑅 ∈ Ring) → (Scalar‘𝑀) ∈ Ring)
3633fveq2d 6667 . . . . . . 7 ((𝐼𝑉𝑅 ∈ Ring) → (Base‘(Scalar‘𝑀)) = (Base‘𝑅))
3736eleq2d 2901 . . . . . 6 ((𝐼𝑉𝑅 ∈ Ring) → (𝑞 ∈ (Base‘(Scalar‘𝑀)) ↔ 𝑞 ∈ (Base‘𝑅)))
3836eleq2d 2901 . . . . . 6 ((𝐼𝑉𝑅 ∈ Ring) → (𝑟 ∈ (Base‘(Scalar‘𝑀)) ↔ 𝑟 ∈ (Base‘𝑅)))
3937, 38anbi12d 633 . . . . 5 ((𝐼𝑉𝑅 ∈ Ring) → ((𝑞 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑀))) ↔ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))))
40 simpll 766 . . . . . . . . . 10 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝐼𝑉)
41 simplr 768 . . . . . . . . . 10 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring)
42 simprr 772 . . . . . . . . . 10 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝑟 ∈ (Base‘𝑅))
4340, 41, 423jca 1125 . . . . . . . . 9 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝐼𝑉𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)))
444lmod1lem1 44849 . . . . . . . . 9 ((𝐼𝑉𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑟( ·𝑠𝑀)𝐼) ∈ {𝐼})
4543, 44syl 17 . . . . . . . 8 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑟( ·𝑠𝑀)𝐼) ∈ {𝐼})
464lmod1lem2 44850 . . . . . . . . 9 ((𝐼𝑉𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑟( ·𝑠𝑀)(𝐼(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)))
4743, 46syl 17 . . . . . . . 8 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝑟( ·𝑠𝑀)(𝐼(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)))
484lmod1lem3 44851 . . . . . . . 8 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = ((𝑞( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)))
4945, 47, 483jca 1125 . . . . . . 7 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑟( ·𝑠𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝐼(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = ((𝑞( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼))))
504lmod1lem4 44852 . . . . . . 7 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝐼)))
514lmod1lem5 44853 . . . . . . . 8 ((𝐼𝑉𝑅 ∈ Ring) → ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼)
5251adantr 484 . . . . . . 7 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼)
5349, 50, 52jca32 519 . . . . . 6 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (((𝑟( ·𝑠𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝐼(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = ((𝑞( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼)))
5453ex 416 . . . . 5 ((𝐼𝑉𝑅 ∈ Ring) → ((𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅)) → (((𝑟( ·𝑠𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝐼(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = ((𝑞( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼))))
5539, 54sylbid 243 . . . 4 ((𝐼𝑉𝑅 ∈ Ring) → ((𝑞 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑀))) → (((𝑟( ·𝑠𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝐼(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = ((𝑞( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼))))
5655ralrimivv 3185 . . 3 ((𝐼𝑉𝑅 ∈ Ring) → ∀𝑞 ∈ (Base‘(Scalar‘𝑀))∀𝑟 ∈ (Base‘(Scalar‘𝑀))(((𝑟( ·𝑠𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝐼(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = ((𝑞( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼)))
57 oveq2 7159 . . . . . . . . . . . 12 (𝑥 = 𝐼 → (𝑤(+g𝑀)𝑥) = (𝑤(+g𝑀)𝐼))
5857oveq2d 7167 . . . . . . . . . . 11 (𝑥 = 𝐼 → (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝑥)) = (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝐼)))
59 oveq2 7159 . . . . . . . . . . . 12 (𝑥 = 𝐼 → (𝑟( ·𝑠𝑀)𝑥) = (𝑟( ·𝑠𝑀)𝐼))
6059oveq2d 7167 . . . . . . . . . . 11 (𝑥 = 𝐼 → ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑥)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)))
6158, 60eqeq12d 2840 . . . . . . . . . 10 (𝑥 = 𝐼 → ((𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝑥)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑥)) ↔ (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼))))
62613anbi2d 1438 . . . . . . . . 9 (𝑥 = 𝐼 → (((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝑥)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑥)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤))) ↔ ((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤)))))
6362anbi1d 632 . . . . . . . 8 (𝑥 = 𝐼 → ((((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝑥)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑥)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = 𝑤)) ↔ (((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = 𝑤))))
6463ralbidv 3192 . . . . . . 7 (𝑥 = 𝐼 → (∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝑥)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑥)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = 𝑤)) ↔ ∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = 𝑤))))
6564ralsng 4598 . . . . . 6 (𝐼𝑉 → (∀𝑥 ∈ {𝐼}∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝑥)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑥)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = 𝑤)) ↔ ∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = 𝑤))))
6665adantr 484 . . . . 5 ((𝐼𝑉𝑅 ∈ Ring) → (∀𝑥 ∈ {𝐼}∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝑥)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑥)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = 𝑤)) ↔ ∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = 𝑤))))
67 oveq2 7159 . . . . . . . . . 10 (𝑤 = 𝐼 → (𝑟( ·𝑠𝑀)𝑤) = (𝑟( ·𝑠𝑀)𝐼))
6867eleq1d 2900 . . . . . . . . 9 (𝑤 = 𝐼 → ((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ↔ (𝑟( ·𝑠𝑀)𝐼) ∈ {𝐼}))
69 oveq1 7158 . . . . . . . . . . 11 (𝑤 = 𝐼 → (𝑤(+g𝑀)𝐼) = (𝐼(+g𝑀)𝐼))
7069oveq2d 7167 . . . . . . . . . 10 (𝑤 = 𝐼 → (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝐼)) = (𝑟( ·𝑠𝑀)(𝐼(+g𝑀)𝐼)))
7167oveq1d 7166 . . . . . . . . . 10 (𝑤 = 𝐼 → ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)))
7270, 71eqeq12d 2840 . . . . . . . . 9 (𝑤 = 𝐼 → ((𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ↔ (𝑟( ·𝑠𝑀)(𝐼(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼))))
73 oveq2 7159 . . . . . . . . . 10 (𝑤 = 𝐼 → ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼))
74 oveq2 7159 . . . . . . . . . . 11 (𝑤 = 𝐼 → (𝑞( ·𝑠𝑀)𝑤) = (𝑞( ·𝑠𝑀)𝐼))
7574, 67oveq12d 7169 . . . . . . . . . 10 (𝑤 = 𝐼 → ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤)) = ((𝑞( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)))
7673, 75eqeq12d 2840 . . . . . . . . 9 (𝑤 = 𝐼 → (((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤)) ↔ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = ((𝑞( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼))))
7768, 72, 763anbi123d 1433 . . . . . . . 8 (𝑤 = 𝐼 → (((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤))) ↔ ((𝑟( ·𝑠𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝐼(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = ((𝑞( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)))))
78 oveq2 7159 . . . . . . . . . 10 (𝑤 = 𝐼 → ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼))
7967oveq2d 7167 . . . . . . . . . 10 (𝑤 = 𝐼 → (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝑤)) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝐼)))
8078, 79eqeq12d 2840 . . . . . . . . 9 (𝑤 = 𝐼 → (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝑤)) ↔ ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝐼))))
81 oveq2 7159 . . . . . . . . . 10 (𝑤 = 𝐼 → ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼))
82 id 22 . . . . . . . . . 10 (𝑤 = 𝐼𝑤 = 𝐼)
8381, 82eqeq12d 2840 . . . . . . . . 9 (𝑤 = 𝐼 → (((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = 𝑤 ↔ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼))
8480, 83anbi12d 633 . . . . . . . 8 (𝑤 = 𝐼 → ((((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = 𝑤) ↔ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼)))
8577, 84anbi12d 633 . . . . . . 7 (𝑤 = 𝐼 → ((((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = 𝑤)) ↔ (((𝑟( ·𝑠𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝐼(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = ((𝑞( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼))))
8685ralsng 4598 . . . . . 6 (𝐼𝑉 → (∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = 𝑤)) ↔ (((𝑟( ·𝑠𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝐼(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = ((𝑞( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼))))
8786adantr 484 . . . . 5 ((𝐼𝑉𝑅 ∈ Ring) → (∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = 𝑤)) ↔ (((𝑟( ·𝑠𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝐼(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = ((𝑞( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼))))
8866, 87bitrd 282 . . . 4 ((𝐼𝑉𝑅 ∈ Ring) → (∀𝑥 ∈ {𝐼}∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝑥)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑥)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = 𝑤)) ↔ (((𝑟( ·𝑠𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝐼(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = ((𝑞( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼))))
89882ralbidv 3194 . . 3 ((𝐼𝑉𝑅 ∈ Ring) → (∀𝑞 ∈ (Base‘(Scalar‘𝑀))∀𝑟 ∈ (Base‘(Scalar‘𝑀))∀𝑥 ∈ {𝐼}∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝑥)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑥)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = 𝑤)) ↔ ∀𝑞 ∈ (Base‘(Scalar‘𝑀))∀𝑟 ∈ (Base‘(Scalar‘𝑀))(((𝑟( ·𝑠𝑀)𝐼) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝐼(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = ((𝑞( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝐼)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼))))
9056, 89mpbird 260 . 2 ((𝐼𝑉𝑅 ∈ Ring) → ∀𝑞 ∈ (Base‘(Scalar‘𝑀))∀𝑟 ∈ (Base‘(Scalar‘𝑀))∀𝑥 ∈ {𝐼}∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝑥)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑥)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = 𝑤)))
914lmodbase 16639 . . . 4 ({𝐼} ∈ V → {𝐼} = (Base‘𝑀))
925, 91ax-mp 5 . . 3 {𝐼} = (Base‘𝑀)
93 eqid 2824 . . 3 (+g𝑀) = (+g𝑀)
94 eqid 2824 . . 3 ( ·𝑠𝑀) = ( ·𝑠𝑀)
95 eqid 2824 . . 3 (Scalar‘𝑀) = (Scalar‘𝑀)
96 eqid 2824 . . 3 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
97 eqid 2824 . . 3 (+g‘(Scalar‘𝑀)) = (+g‘(Scalar‘𝑀))
98 eqid 2824 . . 3 (.r‘(Scalar‘𝑀)) = (.r‘(Scalar‘𝑀))
99 eqid 2824 . . 3 (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
10092, 93, 94, 95, 96, 97, 98, 99islmod 19640 . 2 (𝑀 ∈ LMod ↔ (𝑀 ∈ Grp ∧ (Scalar‘𝑀) ∈ Ring ∧ ∀𝑞 ∈ (Base‘(Scalar‘𝑀))∀𝑟 ∈ (Base‘(Scalar‘𝑀))∀𝑥 ∈ {𝐼}∀𝑤 ∈ {𝐼} (((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ∧ (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝑥)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑥)) ∧ ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤))) ∧ (((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝑤)) ∧ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = 𝑤))))
10130, 35, 90, 100syl3anbrc 1340 1 ((𝐼𝑉𝑅 ∈ Ring) → 𝑀 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  Vcvv 3480  cun 3917  {csn 4550  {cpr 4552  {ctp 4554  cop 4556  cfv 6345  (class class class)co 7151  cmpo 7153  ndxcnx 16482  Basecbs 16485  +gcplusg 16567  .rcmulr 16568  Scalarcsca 16570   ·𝑠 cvsca 16571  Grpcgrp 18105  1rcur 19253  Ringcrg 19299  LModclmod 19636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7577  df-1st 7686  df-2nd 7687  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-1o 8100  df-oadd 8104  df-er 8287  df-en 8508  df-dom 8509  df-sdom 8510  df-fin 8511  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11637  df-2 11699  df-3 11700  df-4 11701  df-5 11702  df-6 11703  df-n0 11897  df-z 11981  df-uz 12243  df-fz 12897  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-plusg 16580  df-sca 16583  df-vsca 16584  df-0g 16717  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-grp 18108  df-mgp 19242  df-ur 19254  df-ring 19301  df-lmod 19638
This theorem is referenced by:  lmod1zr  44855
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