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Theorem lmod1 43076
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 2799 . . . . 5 {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}
21grp1 17838 . . . 4 (𝐼𝑉 → {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} ∈ Grp)
3 fvex 6424 . . . . . . 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 5099 . . . . . . . . . . . . 13 {𝐼} ∈ V
61grpbase 16312 . . . . . . . . . . . . 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 4597 . . . . . . . . . . 11 ⟨(Base‘ndx), {𝐼}⟩ = ⟨(Base‘ndx), (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩
9 tpeq1 4466 . . . . . . . . . . 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 3961 . . . . . . . . 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 2821 . . . . . . . 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 16339 . . . . . . 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 2808 . . . . 5 (Base‘𝑀) = (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})
16 fvex 6424 . . . . . . 7 (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) ∈ V
17 snex 5099 . . . . . . . . . . . . 13 {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} ∈ V
181grpplusg 16313 . . . . . . . . . . . . 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 4597 . . . . . . . . . . 11 ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩ = ⟨(+g‘ndx), (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩
21 tpeq2 4467 . . . . . . . . . . 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 3961 . . . . . . . . 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 2821 . . . . . . . 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 16340 . . . . . . 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 2808 . . . . 5 (+g𝑀) = (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})
2815, 27grpprop 17754 . . . 4 (𝑀 ∈ Grp ↔ {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} ∈ Grp)
292, 28sylibr 226 . . 3 (𝐼𝑉𝑀 ∈ Grp)
3029adantr 473 . 2 ((𝐼𝑉𝑅 ∈ Ring) → 𝑀 ∈ Grp)
314lmodsca 16341 . . . . 5 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑀))
3231eqcomd 2805 . . . 4 (𝑅 ∈ Ring → (Scalar‘𝑀) = 𝑅)
3332adantl 474 . . 3 ((𝐼𝑉𝑅 ∈ Ring) → (Scalar‘𝑀) = 𝑅)
34 simpr 478 . . 3 ((𝐼𝑉𝑅 ∈ Ring) → 𝑅 ∈ Ring)
3533, 34eqeltrd 2878 . 2 ((𝐼𝑉𝑅 ∈ Ring) → (Scalar‘𝑀) ∈ Ring)
3633fveq2d 6415 . . . . . . 7 ((𝐼𝑉𝑅 ∈ Ring) → (Base‘(Scalar‘𝑀)) = (Base‘𝑅))
3736eleq2d 2864 . . . . . 6 ((𝐼𝑉𝑅 ∈ Ring) → (𝑞 ∈ (Base‘(Scalar‘𝑀)) ↔ 𝑞 ∈ (Base‘𝑅)))
3836eleq2d 2864 . . . . . 6 ((𝐼𝑉𝑅 ∈ Ring) → (𝑟 ∈ (Base‘(Scalar‘𝑀)) ↔ 𝑟 ∈ (Base‘𝑅)))
3937, 38anbi12d 625 . . . . 5 ((𝐼𝑉𝑅 ∈ Ring) → ((𝑞 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑀))) ↔ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))))
40 simpll 784 . . . . . . . . . 10 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝐼𝑉)
41 simplr 786 . . . . . . . . . 10 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring)
42 simprr 790 . . . . . . . . . 10 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝑟 ∈ (Base‘𝑅))
4340, 41, 423jca 1159 . . . . . . . . 9 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝐼𝑉𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)))
444lmod1lem1 43071 . . . . . . . . 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 43072 . . . . . . . . 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 43073 . . . . . . . 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 1159 . . . . . . 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 43074 . . . . . . 7 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝐼)))
514lmod1lem5 43075 . . . . . . . 8 ((𝐼𝑉𝑅 ∈ Ring) → ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼)
5251adantr 473 . . . . . . 7 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼)
5349, 50, 52jca32 512 . . . . . 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 402 . . . . 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 232 . . . 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 3151 . . 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 6886 . . . . . . . . . . . 12 (𝑥 = 𝐼 → (𝑤(+g𝑀)𝑥) = (𝑤(+g𝑀)𝐼))
5857oveq2d 6894 . . . . . . . . . . 11 (𝑥 = 𝐼 → (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝑥)) = (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝐼)))
59 oveq2 6886 . . . . . . . . . . . 12 (𝑥 = 𝐼 → (𝑟( ·𝑠𝑀)𝑥) = (𝑟( ·𝑠𝑀)𝐼))
6059oveq2d 6894 . . . . . . . . . . 11 (𝑥 = 𝐼 → ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑥)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)))
6158, 60eqeq12d 2814 . . . . . . . . . 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 1566 . . . . . . . . 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 624 . . . . . . . 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 3167 . . . . . . 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 4409 . . . . . 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 473 . . . . 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 6886 . . . . . . . . . 10 (𝑤 = 𝐼 → (𝑟( ·𝑠𝑀)𝑤) = (𝑟( ·𝑠𝑀)𝐼))
6867eleq1d 2863 . . . . . . . . 9 (𝑤 = 𝐼 → ((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ↔ (𝑟( ·𝑠𝑀)𝐼) ∈ {𝐼}))
69 oveq1 6885 . . . . . . . . . . 11 (𝑤 = 𝐼 → (𝑤(+g𝑀)𝐼) = (𝐼(+g𝑀)𝐼))
7069oveq2d 6894 . . . . . . . . . 10 (𝑤 = 𝐼 → (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝐼)) = (𝑟( ·𝑠𝑀)(𝐼(+g𝑀)𝐼)))
7167oveq1d 6893 . . . . . . . . . 10 (𝑤 = 𝐼 → ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)))
7270, 71eqeq12d 2814 . . . . . . . . 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 6886 . . . . . . . . . 10 (𝑤 = 𝐼 → ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼))
74 oveq2 6886 . . . . . . . . . . 11 (𝑤 = 𝐼 → (𝑞( ·𝑠𝑀)𝑤) = (𝑞( ·𝑠𝑀)𝐼))
7574, 67oveq12d 6896 . . . . . . . . . 10 (𝑤 = 𝐼 → ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤)) = ((𝑞( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)))
7673, 75eqeq12d 2814 . . . . . . . . 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 1561 . . . . . . . 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 6886 . . . . . . . . . 10 (𝑤 = 𝐼 → ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼))
7967oveq2d 6894 . . . . . . . . . 10 (𝑤 = 𝐼 → (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝑤)) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝐼)))
8078, 79eqeq12d 2814 . . . . . . . . 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 6886 . . . . . . . . . 10 (𝑤 = 𝐼 → ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼))
82 id 22 . . . . . . . . . 10 (𝑤 = 𝐼𝑤 = 𝐼)
8381, 82eqeq12d 2814 . . . . . . . . 9 (𝑤 = 𝐼 → (((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = 𝑤 ↔ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼))
8480, 83anbi12d 625 . . . . . . . 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 625 . . . . . . 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 4409 . . . . . 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 473 . . . . 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 271 . . . 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 3170 . . 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 249 . 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 16339 . . . 4 ({𝐼} ∈ V → {𝐼} = (Base‘𝑀))
925, 91ax-mp 5 . . 3 {𝐼} = (Base‘𝑀)
93 eqid 2799 . . 3 (+g𝑀) = (+g𝑀)
94 eqid 2799 . . 3 ( ·𝑠𝑀) = ( ·𝑠𝑀)
95 eqid 2799 . . 3 (Scalar‘𝑀) = (Scalar‘𝑀)
96 eqid 2799 . . 3 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
97 eqid 2799 . . 3 (+g‘(Scalar‘𝑀)) = (+g‘(Scalar‘𝑀))
98 eqid 2799 . . 3 (.r‘(Scalar‘𝑀)) = (.r‘(Scalar‘𝑀))
99 eqid 2799 . . 3 (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
10092, 93, 94, 95, 96, 97, 98, 99islmod 19185 . 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 1444 1 ((𝐼𝑉𝑅 ∈ Ring) → 𝑀 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3089  Vcvv 3385  cun 3767  {csn 4368  {cpr 4370  {ctp 4372  cop 4374  cfv 6101  (class class class)co 6878  cmpt2 6880  ndxcnx 16181  Basecbs 16184  +gcplusg 16267  .rcmulr 16268  Scalarcsca 16270   ·𝑠 cvsca 16271  Grpcgrp 17738  1rcur 18817  Ringcrg 18863  LModclmod 19181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-2 11376  df-3 11377  df-4 11378  df-5 11379  df-6 11380  df-n0 11581  df-z 11667  df-uz 11931  df-fz 12581  df-struct 16186  df-ndx 16187  df-slot 16188  df-base 16190  df-sets 16191  df-plusg 16280  df-sca 16283  df-vsca 16284  df-0g 16417  df-mgm 17557  df-sgrp 17599  df-mnd 17610  df-grp 17741  df-mgp 18806  df-ur 18818  df-ring 18865  df-lmod 19183
This theorem is referenced by:  lmod1zr  43077
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