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Theorem List for Metamath Proof Explorer - 18401-18500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsylow1lem5 18401* Lemma for sylow1 18402. Using Lagrange's theorem and the orbit-stabilizer theorem, show that there is a subgroup with size exactly 𝑃𝑁. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝑃𝑁) ∥ (♯‘𝑋))    &    + = (+g𝐺)    &   𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)}    &    = (𝑥𝑋, 𝑦𝑆 ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}    &   (𝜑𝐵𝑆)    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐵) = 𝐵}    &   (𝜑 → (𝑃 pCnt (♯‘[𝐵] )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))       (𝜑 → ∃ ∈ (SubGrp‘𝐺)(♯‘) = (𝑃𝑁))

Theoremsylow1 18402* Sylow's first theorem. If 𝑃𝑁 is a prime power that divides the cardinality of 𝐺, then 𝐺 has a supgroup with size 𝑃𝑁. This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝑃𝑁) ∥ (♯‘𝑋))       (𝜑 → ∃𝑔 ∈ (SubGrp‘𝐺)(♯‘𝑔) = (𝑃𝑁))

Theoremodcau 18403* Cauchy's theorem for the order of an element in a group. A finite group whose order divides a prime 𝑃 contains an element of order 𝑃. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)       (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (♯‘𝑋)) → ∃𝑔𝑋 (𝑂𝑔) = 𝑃)

Theorempgpfi 18404* The converse to pgpfi1 18394. A finite group is a 𝑃-group iff it has size some power of 𝑃. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃𝑛))))

Theorempgpfi2 18405 Alternate version of pgpfi 18404. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝑋 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ (♯‘𝑋) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))))

Theorempgphash 18406 The order of a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝑋 = (Base‘𝐺)       ((𝑃 pGrp 𝐺𝑋 ∈ Fin) → (♯‘𝑋) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))

Theoremisslw 18407* The property of being a Sylow subgroup. A Sylow 𝑃-subgroup is a 𝑃-group which has no proper supersets that are also 𝑃-groups. (Contributed by Mario Carneiro, 16-Jan-2015.)
(𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻𝑘𝑃 pGrp (𝐺s 𝑘)) ↔ 𝐻 = 𝑘)))

Theoremslwprm 18408 Reverse closure for the first argument of a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.) (Revised by Mario Carneiro, 2-May-2015.)
(𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)

Theoremslwsubg 18409 A Sylow 𝑃-subgroup is a subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
(𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))

Theoremslwispgp 18410 Defining property of a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑆 = (𝐺s 𝐾)       ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻𝐾𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))

Theoremslwpss 18411 A proper superset of a Sylow subgroup is not a 𝑃-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑆 = (𝐺s 𝐾)       ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻𝐾) → ¬ 𝑃 pGrp 𝑆)

Theoremslwpgp 18412 A Sylow 𝑃-subgroup is a 𝑃-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑆 = (𝐺s 𝐻)       (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp 𝑆)

Theorempgpssslw 18413* Every 𝑃-subgroup is contained in a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝑆 = (𝐺s 𝐻)    &   𝐹 = (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} ↦ (♯‘𝑥))       ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑘 ∈ (𝑃 pSyl 𝐺)𝐻𝑘)

Theoremslwn0 18414 Every finite group contains a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) → (𝑃 pSyl 𝐺) ≠ ∅)

Theoremsubgslw 18415 A Sylow subgroup that is contained in a larger subgroup is also Sylow with respect to the subgroup. (The converse need not be true.) (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐻 = (𝐺s 𝑆)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾𝑆) → 𝐾 ∈ (𝑃 pSyl 𝐻))

Theoremsylow2alem1 18416* Lemma for sylow2a 18418. An equivalence class of fixed points is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑 ∈ (𝐺 GrpAct 𝑌))    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑌 ∈ Fin)    &   𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}       ((𝜑𝐴𝑍) → [𝐴] = {𝐴})

Theoremsylow2alem2 18417* Lemma for sylow2a 18418. All the orbits which are not for fixed points have size 𝐺 ∣ / ∣ 𝐺𝑥 (where 𝐺𝑥 is the stabilizer subgroup) and thus are powers of 𝑃. And since they are all nontrivial (because any orbit which is a singleton is a fixed point), they all divide 𝑃, and so does the sum of all of them. (Contributed by Mario Carneiro, 17-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑 ∈ (𝐺 GrpAct 𝑌))    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑌 ∈ Fin)    &   𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}       (𝜑𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ) ∖ 𝒫 𝑍)(♯‘𝑧))

Theoremsylow2a 18418* A named lemma of Sylow's second and third theorems. If 𝐺 is a finite 𝑃-group that acts on the finite set 𝑌, then the set 𝑍 of all points of 𝑌 fixed by every element of 𝐺 has cardinality equivalent to the cardinality of 𝑌, mod 𝑃. (Contributed by Mario Carneiro, 17-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑 ∈ (𝐺 GrpAct 𝑌))    &   (𝜑𝑃 pGrp 𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑌 ∈ Fin)    &   𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}       (𝜑𝑃 ∥ ((♯‘𝑌) − (♯‘𝑍)))

Theoremsylow2blem1 18419* Lemma for sylow2b 18422. Evaluate the group action on a left coset. (Contributed by Mario Carneiro, 17-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐻 ∈ (SubGrp‘𝐺))    &   (𝜑𝐾 ∈ (SubGrp‘𝐺))    &    + = (+g𝐺)    &    = (𝐺 ~QG 𝐾)    &    · = (𝑥𝐻, 𝑦 ∈ (𝑋 / ) ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))       ((𝜑𝐵𝐻𝐶𝑋) → (𝐵 · [𝐶] ) = [(𝐵 + 𝐶)] )

Theoremsylow2blem2 18420* Lemma for sylow2b 18422. Left multiplication in a subgroup 𝐻 is a group action on the set of all left cosets of 𝐾. (Contributed by Mario Carneiro, 17-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐻 ∈ (SubGrp‘𝐺))    &   (𝜑𝐾 ∈ (SubGrp‘𝐺))    &    + = (+g𝐺)    &    = (𝐺 ~QG 𝐾)    &    · = (𝑥𝐻, 𝑦 ∈ (𝑋 / ) ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))       (𝜑· ∈ ((𝐺s 𝐻) GrpAct (𝑋 / )))

Theoremsylow2blem3 18421* Sylow's second theorem. Putting together the results of sylow2a 18418 and the orbit-stabilizer theorem to show that 𝑃 does not divide the set of all fixed points under the group action, we get that there is a fixed point of the group action, so that there is some 𝑔𝑋 with 𝑔𝐾 = 𝑔𝐾 for all 𝐻. This implies that invg(𝑔)𝑔𝐾, so is in the conjugated subgroup 𝑔𝐾invg(𝑔). (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐻 ∈ (SubGrp‘𝐺))    &   (𝜑𝐾 ∈ (SubGrp‘𝐺))    &    + = (+g𝐺)    &    = (𝐺 ~QG 𝐾)    &    · = (𝑥𝐻, 𝑦 ∈ (𝑋 / ) ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))    &   (𝜑𝑃 pGrp (𝐺s 𝐻))    &   (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))    &    = (-g𝐺)       (𝜑 → ∃𝑔𝑋 𝐻 ⊆ ran (𝑥𝐾 ↦ ((𝑔 + 𝑥) 𝑔)))

Theoremsylow2b 18422* Sylow's second theorem. Any 𝑃-group 𝐻 is a subgroup of a conjugated 𝑃-group 𝐾 of order 𝑃𝑛 ∥ (♯‘𝑋) with 𝑛 maximal. This is usually stated under the assumption that 𝐾 is a Sylow subgroup, but we use a slightly different definition, whose equivalence to this one requires this theorem. This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐻 ∈ (SubGrp‘𝐺))    &   (𝜑𝐾 ∈ (SubGrp‘𝐺))    &    + = (+g𝐺)    &   (𝜑𝑃 pGrp (𝐺s 𝐻))    &   (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))    &    = (-g𝐺)       (𝜑 → ∃𝑔𝑋 𝐻 ⊆ ran (𝑥𝐾 ↦ ((𝑔 + 𝑥) 𝑔)))

Theoremslwhash 18423 A sylow subgroup has cardinality equal to the maximum power of 𝑃 dividing the group. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐻 ∈ (𝑃 pSyl 𝐺))       (𝜑 → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))

Theoremfislw 18424 The sylow subgroups of a finite group are exactly the groups which have cardinality equal to the maximum power of 𝑃 dividing the group. (Contributed by Mario Carneiro, 16-Jan-2015.)
𝑋 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))))

Theoremsylow2 18425* Sylow's second theorem. See also sylow2b 18422 for the "hard" part of the proof. Any two Sylow 𝑃-subgroups are conjugate to one another, and hence the same size, namely 𝑃↑(𝑃 pCnt ∣ 𝑋 ∣ ) (see fislw 18424). This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 18-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝐻 ∈ (𝑃 pSyl 𝐺))    &   (𝜑𝐾 ∈ (𝑃 pSyl 𝐺))    &    + = (+g𝐺)    &    = (-g𝐺)       (𝜑 → ∃𝑔𝑋 𝐻 = ran (𝑥𝐾 ↦ ((𝑔 + 𝑥) 𝑔)))

Theoremsylow3lem1 18426* Lemma for sylow3 18432, first part. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &    + = (+g𝐺)    &    = (-g𝐺)    &    = (𝑥𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))       (𝜑 ∈ (𝐺 GrpAct (𝑃 pSyl 𝐺)))

Theoremsylow3lem2 18427* Lemma for sylow3 18432, first part. The stabilizer of a given Sylow subgroup 𝐾 in the group action acting on all of 𝐺 is the normalizer NG(K). (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &    + = (+g𝐺)    &    = (-g𝐺)    &    = (𝑥𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))    &   (𝜑𝐾 ∈ (𝑃 pSyl 𝐺))    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐾) = 𝐾}    &   𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)}       (𝜑𝐻 = 𝑁)

Theoremsylow3lem3 18428* Lemma for sylow3 18432, first part. The number of Sylow subgroups is the same as the index (number of cosets) of the normalizer of the Sylow subgroup 𝐾. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &    + = (+g𝐺)    &    = (-g𝐺)    &    = (𝑥𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))    &   (𝜑𝐾 ∈ (𝑃 pSyl 𝐺))    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐾) = 𝐾}    &   𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)}       (𝜑 → (♯‘(𝑃 pSyl 𝐺)) = (♯‘(𝑋 / (𝐺 ~QG 𝑁))))

Theoremsylow3lem4 18429* Lemma for sylow3 18432, first part. The number of Sylow subgroups is a divisor of the size of 𝐺 reduced by the size of a Sylow subgroup of 𝐺. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &    + = (+g𝐺)    &    = (-g𝐺)    &    = (𝑥𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))    &   (𝜑𝐾 ∈ (𝑃 pSyl 𝐺))    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐾) = 𝐾}    &   𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)}       (𝜑 → (♯‘(𝑃 pSyl 𝐺)) ∥ ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))))

Theoremsylow3lem5 18430* Lemma for sylow3 18432, second part. Reduce the group action of sylow3lem1 18426 to a given Sylow subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &    + = (+g𝐺)    &    = (-g𝐺)    &   (𝜑𝐾 ∈ (𝑃 pSyl 𝐺))    &    = (𝑥𝐾, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))       (𝜑 ∈ ((𝐺s 𝐾) GrpAct (𝑃 pSyl 𝐺)))

Theoremsylow3lem6 18431* Lemma for sylow3 18432, second part. Using the lemma sylow2a 18418, show that the number of sylow subgroups is equivalent mod 𝑃 to the number of fixed points under the group action. But 𝐾 is the unique element of the set of Sylow subgroups that is fixed under the group action, so there is exactly one fixed point and so ((♯‘(𝑃 pSyl 𝐺)) mod 𝑃) = 1. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &    + = (+g𝐺)    &    = (-g𝐺)    &   (𝜑𝐾 ∈ (𝑃 pSyl 𝐺))    &    = (𝑥𝐾, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧𝑦 ↦ ((𝑥 + 𝑧) 𝑥)))    &   𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)}       (𝜑 → ((♯‘(𝑃 pSyl 𝐺)) mod 𝑃) = 1)

Theoremsylow3 18432 Sylow's third theorem. The number of Sylow subgroups is a divisor of 𝐺 ∣ / 𝑑, where 𝑑 is the common order of a Sylow subgroup, and is equivalent to 1 mod 𝑃. This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝑋 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑃 ∈ ℙ)    &   𝑁 = (♯‘(𝑃 pSyl 𝐺))       (𝜑 → (𝑁 ∥ ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))) ∧ (𝑁 mod 𝑃) = 1))

10.2.11  Direct products

Syntaxclsm 18433 Extend class notation with subgroup sum.
class LSSum

Syntaxcpj1 18434 Extend class notation with left projection.
class proj1

Definitiondf-lsm 18435* Define subgroup sum (inner direct product of subgroups). (Contributed by NM, 28-Jan-2014.)
LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝑤)𝑦))))

Definitiondf-pj1 18436* Define the left projection function, which takes two subgroups 𝑡, 𝑢 with trivial intersection and returns a function mapping the elements of the subgroup sum 𝑡 + 𝑢 to their projections onto 𝑡. (The other projection function can be obtained by swapping the roles of 𝑡 and 𝑢.) (Contributed by Mario Carneiro, 15-Oct-2015.)
proj1 = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ (𝑧 ∈ (𝑡(LSSum‘𝑤)𝑢) ↦ (𝑥𝑡𝑦𝑢 𝑧 = (𝑥(+g𝑤)𝑦)))))

Theoremlsmfval 18437* The subgroup sum function (for a group or vector space). (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)       (𝐺𝑉 = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥 + 𝑦))))

Theoremlsmvalx 18438* Subspace sum value (for a group or vector space). Extended domain version of lsmval 18447. (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)       ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))

Theoremlsmelvalx 18439* Subspace sum membership (for a group or vector space). Extended domain version of lsmelval 18448. (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)       ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 + 𝑧)))

Theoremlsmelvalix 18440 Subspace sum membership (for a group or vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)       (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑋𝑇𝑌𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 𝑈))

Theoremoppglsm 18441 The subspace sum operation in the opposite group. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝑂 = (oppg𝐺)    &    = (LSSum‘𝐺)       (𝑇(LSSum‘𝑂)𝑈) = (𝑈 𝑇)

Theoremlsmssv 18442 Subgroup sum is a subset of the base. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)       ((𝐺 ∈ Mnd ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) ⊆ 𝐵)

Theoremlsmless1x 18443 Subset implies subgroup sum subset (extended domain version). (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)       (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → (𝑅 𝑈) ⊆ (𝑇 𝑈))

Theoremlsmless2x 18444 Subset implies subgroup sum subset (extended domain version). (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)       (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (𝑅 𝑇) ⊆ (𝑅 𝑈))

Theoremlsmub1x 18445 Subgroup sum is an upper bound of its arguments. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)       ((𝑇𝐵𝑈 ∈ (SubMnd‘𝐺)) → 𝑇 ⊆ (𝑇 𝑈))

Theoremlsmub2x 18446 Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)       ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈𝐵) → 𝑈 ⊆ (𝑇 𝑈))

Theoremlsmval 18447* Subgroup sum value (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))

Theoremlsmelval 18448* Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 + 𝑧)))

Theoremlsmelvali 18449 Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)       (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑋𝑇𝑌𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 𝑈))

Theoremlsmelvalm 18450* Subgroup sum membership analogue of lsmelval 18448 using vector subtraction. TODO: any way to shorten proof? (Contributed by NM, 16-Mar-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (-g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))       (𝜑 → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 𝑧)))

Theoremlsmelvalmi 18451 Membership of vector subtraction in subgroup sum. (Contributed by NM, 27-Apr-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (-g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑𝑋𝑇)    &   (𝜑𝑌𝑈)       (𝜑 → (𝑋 𝑌) ∈ (𝑇 𝑈))

Theoremlsmsubm 18452 The sum of two commuting submonoids is a submonoid. (Contributed by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)    &   𝑍 = (Cntz‘𝐺)       ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubMnd‘𝐺))

Theoremlsmsubg 18453 The sum of two commuting subgroups is a subgroup. (Contributed by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)    &   𝑍 = (Cntz‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubGrp‘𝐺))

Theoremlsmcom2 18454 Subgroup sum commutes. (Contributed by Mario Carneiro, 22-Apr-2016.)
= (LSSum‘𝐺)    &   𝑍 = (Cntz‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) = (𝑈 𝑇))

Theoremlsmub1 18455 Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑇 ⊆ (𝑇 𝑈))

Theoremlsmub2 18456 Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑈 ⊆ (𝑇 𝑈))

Theoremlsmunss 18457 Union of subgroups is a subset of subgroup sum. (Contributed by NM, 6-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.)
= (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇𝑈) ⊆ (𝑇 𝑈))

Theoremlsmless1 18458 Subset implies subgroup sum subset. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑆𝑇) → (𝑆 𝑈) ⊆ (𝑇 𝑈))

Theoremlsmless2 18459 Subset implies subgroup sum subset. (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑈) → (𝑆 𝑇) ⊆ (𝑆 𝑈))

Theoremlsmless12 18460 Subset implies subgroup sum subset. (Contributed by NM, 14-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅𝑆𝑇𝑈)) → (𝑅 𝑇) ⊆ (𝑆 𝑈))

Theoremlsmidm 18461 Subgroup sum is idempotent. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
= (LSSum‘𝐺)       (𝑈 ∈ (SubGrp‘𝐺) → (𝑈 𝑈) = 𝑈)

Theoremlsmlub 18462 The least upper bound property of subgroup sum. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
= (LSSum‘𝐺)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → ((𝑆𝑈𝑇𝑈) ↔ (𝑆 𝑇) ⊆ 𝑈))

Theoremlsmss1 18463 Subgroup sum with a subset. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑈) → (𝑇 𝑈) = 𝑈)

Theoremlsmss1b 18464 Subgroup sum with a subset. (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇𝑈 ↔ (𝑇 𝑈) = 𝑈))

Theoremlsmss2 18465 Subgroup sum with a subset. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑈𝑇) → (𝑇 𝑈) = 𝑇)

Theoremlsmss2b 18466 Subgroup sum with a subset. (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑈𝑇 ↔ (𝑇 𝑈) = 𝑇))

Theoremlsmass 18467 Subgroup sum is associative. (Contributed by NM, 2-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       ((𝑅 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → ((𝑅 𝑇) 𝑈) = (𝑅 (𝑇 𝑈)))

Theoremlsm01 18468 Subgroup sum with the zero subgroup. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
0 = (0g𝐺)    &    = (LSSum‘𝐺)       (𝑋 ∈ (SubGrp‘𝐺) → (𝑋 { 0 }) = 𝑋)

Theoremlsm02 18469 Subgroup sum with the zero subgroup. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
0 = (0g𝐺)    &    = (LSSum‘𝐺)       (𝑋 ∈ (SubGrp‘𝐺) → ({ 0 } 𝑋) = 𝑋)

Theoremsubglsm 18470 The subgroup sum evaluated within a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐻 = (𝐺s 𝑆)    &    = (LSSum‘𝐺)    &   𝐴 = (LSSum‘𝐻)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → (𝑇 𝑈) = (𝑇𝐴𝑈))

Theoremlssnle 18471 Equivalent expressions for "not less than". (chnlei 28916 analog.) (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))       (𝜑 → (¬ 𝑈𝑇𝑇 ⊊ (𝑇 𝑈)))

Theoremlsmmod 18472 The modular law holds for subgroup sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑆𝑈) → (𝑆 (𝑇𝑈)) = ((𝑆 𝑇) ∩ 𝑈))

Theoremlsmmod2 18473 Modular law dual for subgroup sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 8-Jan-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
= (LSSum‘𝐺)       (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈𝑆) → (𝑆 ∩ (𝑇 𝑈)) = ((𝑆𝑇) 𝑈))

Theoremlsmpropd 18474* If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 29-Jun-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   (𝜑𝐾 ∈ V)    &   (𝜑𝐿 ∈ V)       (𝜑 → (LSSum‘𝐾) = (LSSum‘𝐿))

Theoremcntzrecd 18475 Commute the "subgroups commute" predicate. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ⊆ (𝑍𝑈))       (𝜑𝑈 ⊆ (𝑍𝑇))

Theoremlsmcntz 18476 The "subgroups commute" predicate applied to a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   𝑍 = (Cntz‘𝐺)       (𝜑 → ((𝑆 𝑇) ⊆ (𝑍𝑈) ↔ (𝑆 ⊆ (𝑍𝑈) ∧ 𝑇 ⊆ (𝑍𝑈))))

Theoremlsmcntzr 18477 The "subgroups commute" predicate applied to a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   𝑍 = (Cntz‘𝐺)       (𝜑 → (𝑆 ⊆ (𝑍‘(𝑇 𝑈)) ↔ (𝑆 ⊆ (𝑍𝑇) ∧ 𝑆 ⊆ (𝑍𝑈))))

Theoremlsmdisj 18478 Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &    0 = (0g𝐺)    &   (𝜑 → ((𝑆 𝑇) ∩ 𝑈) = { 0 })       (𝜑 → ((𝑆𝑈) = { 0 } ∧ (𝑇𝑈) = { 0 }))

Theoremlsmdisj2 18479 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &    0 = (0g𝐺)    &   (𝜑 → ((𝑆 𝑇) ∩ 𝑈) = { 0 })    &   (𝜑 → (𝑆𝑇) = { 0 })       (𝜑 → (𝑇 ∩ (𝑆 𝑈)) = { 0 })

Theoremlsmdisj3 18480 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &    0 = (0g𝐺)    &   (𝜑 → ((𝑆 𝑇) ∩ 𝑈) = { 0 })    &   (𝜑 → (𝑆𝑇) = { 0 })    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑆 ⊆ (𝑍𝑇))       (𝜑 → (𝑆 ∩ (𝑇 𝑈)) = { 0 })

Theoremlsmdisjr 18481 Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &    0 = (0g𝐺)    &   (𝜑 → (𝑆 ∩ (𝑇 𝑈)) = { 0 })       (𝜑 → ((𝑆𝑇) = { 0 } ∧ (𝑆𝑈) = { 0 }))

Theoremlsmdisj2r 18482 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &    0 = (0g𝐺)    &   (𝜑 → (𝑆 ∩ (𝑇 𝑈)) = { 0 })    &   (𝜑 → (𝑇𝑈) = { 0 })       (𝜑 → ((𝑆 𝑈) ∩ 𝑇) = { 0 })

Theoremlsmdisj3r 18483 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &    0 = (0g𝐺)    &   (𝜑 → (𝑆 ∩ (𝑇 𝑈)) = { 0 })    &   (𝜑 → (𝑇𝑈) = { 0 })    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ⊆ (𝑍𝑈))       (𝜑 → ((𝑆 𝑇) ∩ 𝑈) = { 0 })

Theoremlsmdisj2a 18484 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &    0 = (0g𝐺)       (𝜑 → ((((𝑆 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆𝑇) = { 0 }) ↔ ((𝑇 ∩ (𝑆 𝑈)) = { 0 } ∧ (𝑆𝑈) = { 0 })))

Theoremlsmdisj2b 18485 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &    0 = (0g𝐺)       (𝜑 → ((((𝑆 𝑈) ∩ 𝑇) = { 0 } ∧ (𝑆𝑈) = { 0 }) ↔ ((𝑆 ∩ (𝑇 𝑈)) = { 0 } ∧ (𝑇𝑈) = { 0 })))

Theoremlsmdisj3a 18486 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑆 ⊆ (𝑍𝑇))       (𝜑 → ((((𝑆 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆𝑇) = { 0 }) ↔ ((𝑆 ∩ (𝑇 𝑈)) = { 0 } ∧ (𝑇𝑈) = { 0 })))

Theoremlsmdisj3b 18487 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ⊆ (𝑍𝑈))       (𝜑 → ((((𝑆 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆𝑇) = { 0 }) ↔ ((𝑆 ∩ (𝑇 𝑈)) = { 0 } ∧ (𝑇𝑈) = { 0 })))

Theoremsubgdisj1 18488 Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. (Contributed by NM, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
+ = (+g𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   (𝜑𝐴𝑇)    &   (𝜑𝐶𝑇)    &   (𝜑𝐵𝑈)    &   (𝜑𝐷𝑈)    &   (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷))       (𝜑𝐴 = 𝐶)

Theoremsubgdisj2 18489 Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. (Contributed by NM, 12-Jul-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
+ = (+g𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   (𝜑𝐴𝑇)    &   (𝜑𝐶𝑇)    &   (𝜑𝐵𝑈)    &   (𝜑𝐷𝑈)    &   (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷))       (𝜑𝐵 = 𝐷)

Theoremsubgdisjb 18490 Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. Analogous to opth 5176, this theorem shows a way of representing a pair of vectors. (Contributed by NM, 5-Jul-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
+ = (+g𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   (𝜑𝐴𝑇)    &   (𝜑𝐶𝑇)    &   (𝜑𝐵𝑈)    &   (𝜑𝐷𝑈)       (𝜑 → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theorempj1fval 18491* The left projection function (for a direct product of group subspaces). (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)    &   𝑃 = (proj1𝐺)       ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 𝑈) ↦ (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦))))

Theorempj1val 18492* The left projection function (for a direct product of group subspaces). (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)    &   𝑃 = (proj1𝐺)       (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) = (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)))

Theorempj1eu 18493* Uniqueness of a left projection. (Contributed by Mario Carneiro, 15-Oct-2015.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))       ((𝜑𝑋 ∈ (𝑇 𝑈)) → ∃!𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦))

Theorempj1f 18494 The left projection function maps a direct subspace sum onto the left factor. (Contributed by Mario Carneiro, 15-Oct-2015.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   𝑃 = (proj1𝐺)       (𝜑 → (𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇)

Theorempj2f 18495 The right projection function maps a direct subspace sum onto the right factor. (Contributed by Mario Carneiro, 15-Oct-2015.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   𝑃 = (proj1𝐺)       (𝜑 → (𝑈𝑃𝑇):(𝑇 𝑈)⟶𝑈)

Theorempj1id 18496 Any element of a direct subspace sum can be decomposed into projections onto the left and right factors. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   𝑃 = (proj1𝐺)       ((𝜑𝑋 ∈ (𝑇 𝑈)) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))

Theorempj1eq 18497 Any element of a direct subspace sum can be decomposed uniquely into projections onto the left and right factors. (Contributed by Mario Carneiro, 16-Oct-2015.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   𝑃 = (proj1𝐺)    &   (𝜑𝑋 ∈ (𝑇 𝑈))    &   (𝜑𝐵𝑇)    &   (𝜑𝐶𝑈)       (𝜑 → (𝑋 = (𝐵 + 𝐶) ↔ (((𝑇𝑃𝑈)‘𝑋) = 𝐵 ∧ ((𝑈𝑃𝑇)‘𝑋) = 𝐶)))

Theorempj1lid 18498 The left projection function is the identity on the left subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   𝑃 = (proj1𝐺)       ((𝜑𝑋𝑇) → ((𝑇𝑃𝑈)‘𝑋) = 𝑋)

Theorempj1rid 18499 The left projection function is the zero operator on the right subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   𝑃 = (proj1𝐺)       ((𝜑𝑋𝑈) → ((𝑇𝑃𝑈)‘𝑋) = 0 )

Theorempj1ghm 18500 The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   𝑃 = (proj1𝐺)       (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺s (𝑇 𝑈)) GrpHom 𝐺))

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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