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Theorem lmod1 42606
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 2651 . . . . 5 {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}
21grp1 17569 . . . 4 (𝐼𝑉 → {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} ∈ Grp)
3 fvex 6239 . . . . . . 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 4938 . . . . . . . . . . . . 13 {𝐼} ∈ V
61grpbase 16038 . . . . . . . . . . . . 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 4437 . . . . . . . . . . 11 ⟨(Base‘ndx), {𝐼}⟩ = ⟨(Base‘ndx), (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩
9 tpeq1 4309 . . . . . . . . . . 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 3796 . . . . . . . . 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 2673 . . . . . . . 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 16065 . . . . . . 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 2660 . . . . 5 (Base‘𝑀) = (Base‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})
16 fvex 6239 . . . . . . 7 (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}) ∈ V
17 snex 4938 . . . . . . . . . . . . 13 {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} ∈ V
181grpplusg 16039 . . . . . . . . . . . . 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 4437 . . . . . . . . . . 11 ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩ = ⟨(+g‘ndx), (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})⟩
21 tpeq2 4310 . . . . . . . . . . 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 3796 . . . . . . . . 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 2673 . . . . . . . 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 16066 . . . . . . 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 2660 . . . . 5 (+g𝑀) = (+g‘{⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩})
2815, 27grpprop 17485 . . . 4 (𝑀 ∈ Grp ↔ {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩} ∈ Grp)
292, 28sylibr 224 . . 3 (𝐼𝑉𝑀 ∈ Grp)
3029adantr 480 . 2 ((𝐼𝑉𝑅 ∈ Ring) → 𝑀 ∈ Grp)
314lmodsca 16067 . . . . 5 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑀))
3231eqcomd 2657 . . . 4 (𝑅 ∈ Ring → (Scalar‘𝑀) = 𝑅)
3332adantl 481 . . 3 ((𝐼𝑉𝑅 ∈ Ring) → (Scalar‘𝑀) = 𝑅)
34 simpr 476 . . 3 ((𝐼𝑉𝑅 ∈ Ring) → 𝑅 ∈ Ring)
3533, 34eqeltrd 2730 . 2 ((𝐼𝑉𝑅 ∈ Ring) → (Scalar‘𝑀) ∈ Ring)
3633fveq2d 6233 . . . . . . 7 ((𝐼𝑉𝑅 ∈ Ring) → (Base‘(Scalar‘𝑀)) = (Base‘𝑅))
3736eleq2d 2716 . . . . . 6 ((𝐼𝑉𝑅 ∈ Ring) → (𝑞 ∈ (Base‘(Scalar‘𝑀)) ↔ 𝑞 ∈ (Base‘𝑅)))
3836eleq2d 2716 . . . . . 6 ((𝐼𝑉𝑅 ∈ Ring) → (𝑟 ∈ (Base‘(Scalar‘𝑀)) ↔ 𝑟 ∈ (Base‘𝑅)))
3937, 38anbi12d 747 . . . . 5 ((𝐼𝑉𝑅 ∈ Ring) → ((𝑞 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑀))) ↔ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))))
40 simpll 805 . . . . . . . . . 10 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝐼𝑉)
41 simplr 807 . . . . . . . . . 10 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring)
42 simprr 811 . . . . . . . . . 10 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → 𝑟 ∈ (Base‘𝑅))
4340, 41, 423jca 1261 . . . . . . . . 9 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → (𝐼𝑉𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)))
444lmod1lem1 42601 . . . . . . . . 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 42602 . . . . . . . . 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 42603 . . . . . . . 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 1261 . . . . . . 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 42604 . . . . . . 7 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝐼)))
514lmod1lem5 42605 . . . . . . . 8 ((𝐼𝑉𝑅 ∈ Ring) → ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼)
5251adantr 480 . . . . . . 7 (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼)
5349, 50, 52jca32 557 . . . . . 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 449 . . . . 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 230 . . . 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 2999 . . 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 6698 . . . . . . . . . . . 12 (𝑥 = 𝐼 → (𝑤(+g𝑀)𝑥) = (𝑤(+g𝑀)𝐼))
5857oveq2d 6706 . . . . . . . . . . 11 (𝑥 = 𝐼 → (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝑥)) = (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝐼)))
59 oveq2 6698 . . . . . . . . . . . 12 (𝑥 = 𝐼 → (𝑟( ·𝑠𝑀)𝑥) = (𝑟( ·𝑠𝑀)𝐼))
6059oveq2d 6706 . . . . . . . . . . 11 (𝑥 = 𝐼 → ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑥)) = ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)))
6158, 60eqeq12d 2666 . . . . . . . . . 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 1444 . . . . . . . . 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 741 . . . . . . . 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 3015 . . . . . . 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 4250 . . . . . 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 480 . . . . 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 6698 . . . . . . . . . 10 (𝑤 = 𝐼 → (𝑟( ·𝑠𝑀)𝑤) = (𝑟( ·𝑠𝑀)𝐼))
6867eleq1d 2715 . . . . . . . . 9 (𝑤 = 𝐼 → ((𝑟( ·𝑠𝑀)𝑤) ∈ {𝐼} ↔ (𝑟( ·𝑠𝑀)𝐼) ∈ {𝐼}))
69 oveq1 6697 . . . . . . . . . . 11 (𝑤 = 𝐼 → (𝑤(+g𝑀)𝐼) = (𝐼(+g𝑀)𝐼))
7069oveq2d 6706 . . . . . . . . . 10 (𝑤 = 𝐼 → (𝑟( ·𝑠𝑀)(𝑤(+g𝑀)𝐼)) = (𝑟( ·𝑠𝑀)(𝐼(+g𝑀)𝐼)))
7167oveq1d 6705 . . . . . . . . . 10 (𝑤 = 𝐼 → ((𝑟( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)))
7270, 71eqeq12d 2666 . . . . . . . . 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 6698 . . . . . . . . . 10 (𝑤 = 𝐼 → ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼))
74 oveq2 6698 . . . . . . . . . . 11 (𝑤 = 𝐼 → (𝑞( ·𝑠𝑀)𝑤) = (𝑞( ·𝑠𝑀)𝐼))
7574, 67oveq12d 6708 . . . . . . . . . 10 (𝑤 = 𝐼 → ((𝑞( ·𝑠𝑀)𝑤)(+g𝑀)(𝑟( ·𝑠𝑀)𝑤)) = ((𝑞( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)))
7673, 75eqeq12d 2666 . . . . . . . . 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 1439 . . . . . . . 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 6698 . . . . . . . . . 10 (𝑤 = 𝐼 → ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝑤) = ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼))
7967oveq2d 6706 . . . . . . . . . 10 (𝑤 = 𝐼 → (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝑤)) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝐼)))
8078, 79eqeq12d 2666 . . . . . . . . 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 6698 . . . . . . . . . 10 (𝑤 = 𝐼 → ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼))
82 id 22 . . . . . . . . . 10 (𝑤 = 𝐼𝑤 = 𝐼)
8381, 82eqeq12d 2666 . . . . . . . . 9 (𝑤 = 𝐼 → (((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝑤) = 𝑤 ↔ ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼))
8480, 83anbi12d 747 . . . . . . . 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 747 . . . . . . 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 4250 . . . . . 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 480 . . . . 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 268 . . . 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 3018 . . 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 247 . 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 16065 . . . 4 ({𝐼} ∈ V → {𝐼} = (Base‘𝑀))
925, 91ax-mp 5 . . 3 {𝐼} = (Base‘𝑀)
93 eqid 2651 . . 3 (+g𝑀) = (+g𝑀)
94 eqid 2651 . . 3 ( ·𝑠𝑀) = ( ·𝑠𝑀)
95 eqid 2651 . . 3 (Scalar‘𝑀) = (Scalar‘𝑀)
96 eqid 2651 . . 3 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
97 eqid 2651 . . 3 (+g‘(Scalar‘𝑀)) = (+g‘(Scalar‘𝑀))
98 eqid 2651 . . 3 (.r‘(Scalar‘𝑀)) = (.r‘(Scalar‘𝑀))
99 eqid 2651 . . 3 (1r‘(Scalar‘𝑀)) = (1r‘(Scalar‘𝑀))
10092, 93, 94, 95, 96, 97, 98, 99islmod 18915 . 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 1265 1 ((𝐼𝑉𝑅 ∈ Ring) → 𝑀 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cun 3605  {csn 4210  {cpr 4212  {ctp 4214  cop 4216  cfv 5926  (class class class)co 6690  cmpt2 6692  ndxcnx 15901  Basecbs 15904  +gcplusg 15988  .rcmulr 15989  Scalarcsca 15991   ·𝑠 cvsca 15992  Grpcgrp 17469  1rcur 18547  Ringcrg 18593  LModclmod 18911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-plusg 16001  df-sca 16004  df-vsca 16005  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-mgp 18536  df-ur 18548  df-ring 18595  df-lmod 18913
This theorem is referenced by:  lmod1zr  42607
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