P. 195 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 19401-19500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pgpfi 19401*	The converse to pgpfi1 19391. 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) = (𝑃↑𝑛))))
Theorem	pgpfi2 19402	Alternate version of pgpfi 19401. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ (♯‘𝑋) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))))
Theorem	pgphash 19403	The order of a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
Theorem	isslw 19404*	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)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)))
Theorem	slwprm 19405	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)
Theorem	slwsubg 19406	A Sylow 𝑃-subgroup is a subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))
Theorem	slwispgp 19407	Defining property of a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ 𝑆 = (𝐺 ↾s 𝐾)    ⇒   ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))
Theorem	slwpss 19408	A proper superset of a Sylow subgroup is not a 𝑃-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ 𝑆 = (𝐺 ↾s 𝐾)    ⇒   ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → ¬ 𝑃 pGrp 𝑆)
Theorem	slwpgp 19409	A Sylow 𝑃-subgroup is a 𝑃-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ 𝑆 = (𝐺 ↾s 𝐻)    ⇒   ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp 𝑆)
Theorem	pgpssslw 19410*	Every 𝑃-subgroup is contained in a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑆 = (𝐺 ↾s 𝐻)    &   ⊢ 𝐹 = (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾s 𝑦) ∧ 𝐻 ⊆ 𝑦)} ↦ (♯‘𝑥))    ⇒   ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑘 ∈ (𝑃 pSyl 𝐺)𝐻 ⊆ 𝑘)
Theorem	slwn0 19411	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) ≠ ∅)
Theorem	subgslw 19412	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 = (𝐺 ↾s 𝑆)    ⇒   ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → 𝐾 ∈ (𝑃 pSyl 𝐻))
Theorem	sylow2alem1 19413*	Lemma for sylow2a 19415. An equivalence class of fixed points is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑌))    &   ⊢ (𝜑 → 𝑃 pGrp 𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑌 ∈ Fin)    &   ⊢ 𝑍 = {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢}    &   ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → [𝐴] ∼ = {𝐴})
Theorem	sylow2alem2 19414*	Lemma for sylow2a 19415. 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))    &   ⊢ (𝜑 → 𝑃 pGrp 𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑌 ∈ Fin)    &   ⊢ 𝑍 = {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢}    &   ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}    ⇒   ⊢ (𝜑 → 𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧))
Theorem	sylow2a 19415*	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))    &   ⊢ (𝜑 → 𝑃 pGrp 𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑌 ∈ Fin)    &   ⊢ 𝑍 = {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢}    &   ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}    ⇒   ⊢ (𝜑 → 𝑃 ∥ ((♯‘𝑌) − (♯‘𝑍)))
Theorem	sylow2blem1 19416*	Lemma for sylow2b 19419. Evaluate the group action on a left coset. (Contributed by Mario Carneiro, 17-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))    &   ⊢ + = (+g‘𝐺)    &   ⊢ ∼ = (𝐺 ~QG 𝐾)    &   ⊢ · = (𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝐵 · [𝐶] ∼ ) = [(𝐵 + 𝐶)] ∼ )
Theorem	sylow2blem2 19417*	Lemma for sylow2b 19419. 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)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))    &   ⊢ + = (+g‘𝐺)    &   ⊢ ∼ = (𝐺 ~QG 𝐾)    &   ⊢ · = (𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))    ⇒   ⊢ (𝜑 → · ∈ ((𝐺 ↾s 𝐻) GrpAct (𝑋 / ∼ )))
Theorem	sylow2blem3 19418*	Sylow's second theorem. Putting together the results of sylow2a 19415 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 𝑔 ∈ 𝑋 with ℎ𝑔𝐾 = 𝑔𝐾 for all ℎ ∈ 𝐻. This implies that invg(𝑔)ℎ𝑔 ∈ 𝐾, so ℎ is in the conjugated subgroup 𝑔𝐾invg(𝑔). (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))    &   ⊢ + = (+g‘𝐺)    &   ⊢ ∼ = (𝐺 ~QG 𝐾)    &   ⊢ · = (𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))    &   ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾s 𝐻))    &   ⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
Theorem	sylow2b 19419*	Sylow's second theorem. Any 𝑃-group 𝐻 is a subgroup of a conjugated 𝑃-group 𝐾 of order 𝑃↑𝑛 ∥ (♯‘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)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾s 𝐻))    &   ⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
Theorem	slwhash 19420	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)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑃 pSyl 𝐺))    ⇒   ⊢ (𝜑 → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
Theorem	fislw 19421	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))))))
Theorem	sylow2 19422*	Sylow's second theorem. See also sylow2b 19419 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 19421). This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑃 pSyl 𝐺))    &   ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺))    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 = ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
Theorem	sylow3lem1 19423*	Lemma for sylow3 19429, first part. (Contributed by Mario Carneiro, 19-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)))    ⇒   ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct (𝑃 pSyl 𝐺)))
Theorem	sylow3lem2 19424*	Lemma for sylow3 19429, 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)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)))    &   ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺))    &   ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐾) = 𝐾}    &   ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)}    ⇒   ⊢ (𝜑 → 𝐻 = 𝑁)
Theorem	sylow3lem3 19425*	Lemma for sylow3 19429, 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)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)))    &   ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺))    &   ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐾) = 𝐾}    &   ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)}    ⇒   ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) = (♯‘(𝑋 / (𝐺 ~QG 𝑁))))
Theorem	sylow3lem4 19426*	Lemma for sylow3 19429, 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)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)))    &   ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺))    &   ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐾) = 𝐾}    &   ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝐾 ↔ (𝑦 + 𝑥) ∈ 𝐾)}    ⇒   ⊢ (𝜑 → (♯‘(𝑃 pSyl 𝐺)) ∥ ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))))
Theorem	sylow3lem5 19427*	Lemma for sylow3 19429, second part. Reduce the group action of sylow3lem1 19423 to a given Sylow subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺))    &   ⊢ ⊕ = (𝑥 ∈ 𝐾, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)))    ⇒   ⊢ (𝜑 → ⊕ ∈ ((𝐺 ↾s 𝐾) GrpAct (𝑃 pSyl 𝐺)))
Theorem	sylow3lem6 19428*	Lemma for sylow3 19429, second part. Using the lemma sylow2a 19415, 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)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ + = (+g‘𝐺)    &   ⊢ − = (-g‘𝐺)    &   ⊢ (𝜑 → 𝐾 ∈ (𝑃 pSyl 𝐺))    &   ⊢ ⊕ = (𝑥 ∈ 𝐾, 𝑦 ∈ (𝑃 pSyl 𝐺) ↦ ran (𝑧 ∈ 𝑦 ↦ ((𝑥 + 𝑧) − 𝑥)))    &   ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)}    ⇒   ⊢ (𝜑 → ((♯‘(𝑃 pSyl 𝐺)) mod 𝑃) = 1)
Theorem	sylow3 19429	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)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ 𝑁 = (♯‘(𝑃 pSyl 𝐺))    ⇒   ⊢ (𝜑 → (𝑁 ∥ ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))) ∧ (𝑁 mod 𝑃) = 1))
10.2.12  Direct products
Syntax	clsm 19430	Extend class notation with subgroup sum.
class LSSum
Syntax	cpj1 19431	Extend class notation with left projection.
class proj1
Definition	df-lsm 19432*	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 (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦))))
Definition	df-pj1 19433*	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) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥(+g‘𝑤)𝑦)))))
Theorem	lsmfval 19434*	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)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑉 → ⊕ = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥 + 𝑦))))
Theorem	lsmvalx 19435*	Subspace sum value (for a group or vector space). Extended domain version of lsmval 19444. (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦)))
Theorem	lsmelvalx 19436*	Subspace sum membership (for a group or vector space). Extended domain version of lsmelval 19445. (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 + 𝑧)))
Theorem	lsmelvalix 19437	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)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈))
Theorem	oppglsm 19438	The subspace sum operation in the opposite group. (Contributed by Mario Carneiro, 19-Apr-2016.) (Proof shortened by AV, 2-Mar-2024.)
⊢ 𝑂 = (oppg‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ (𝑇(LSSum‘𝑂)𝑈) = (𝑈 ⊕ 𝑇)
Theorem	lsmssv 19439	Subgroup sum is a subset of the base. (Contributed by Mario Carneiro, 19-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇 ⊕ 𝑈) ⊆ 𝐵)
Theorem	lsmless1x 19440	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)    ⇒   ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑅 ⊆ 𝑇) → (𝑅 ⊕ 𝑈) ⊆ (𝑇 ⊕ 𝑈))
Theorem	lsmless2x 19441	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)    ⇒   ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑇 ⊆ 𝑈) → (𝑅 ⊕ 𝑇) ⊆ (𝑅 ⊕ 𝑈))
Theorem	lsmub1x 19442	Subgroup sum is an upper bound of its arguments. (Contributed by Mario Carneiro, 19-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ ((𝑇 ⊆ 𝐵 ∧ 𝑈 ∈ (SubMnd‘𝐺)) → 𝑇 ⊆ (𝑇 ⊕ 𝑈))
Theorem	lsmub2x 19443	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) ∧ 𝑈 ⊆ 𝐵) → 𝑈 ⊆ (𝑇 ⊕ 𝑈))
Theorem	lsmval 19444*	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)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 ⊕ 𝑈) = ran (𝑥 ∈ 𝑇, 𝑦 ∈ 𝑈 ↦ (𝑥 + 𝑦)))
Theorem	lsmelval 19445*	Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 + 𝑧)))
Theorem	lsmelvali 19446	Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 ⊕ 𝑈))
Theorem	lsmelvalm 19447*	Subgroup sum membership analogue of lsmelval 19445 using vector subtraction. TODO: any way to shorten proof? (Contributed by NM, 16-Mar-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ − = (-g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 − 𝑧)))
Theorem	lsmelvalmi 19448	Membership of vector subtraction in subgroup sum. (Contributed by NM, 27-Apr-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ − = (-g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑇)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝑇 ⊕ 𝑈))
Theorem	lsmsubm 19449	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 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubMnd‘𝐺))
Theorem	lsmsubg 19450	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 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubGrp‘𝐺))
Theorem	lsmcom2 19451	Subgroup sum commutes. (Contributed by Mario Carneiro, 22-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    ⇒   ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
10.2.12.1  Direct products (extension)
Theorem	smndlsmidm 19452	The direct product is idempotent for submonoids. (Contributed by AV, 27-Dec-2023.)
⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ (𝑈 ∈ (SubMnd‘𝐺) → (𝑈 ⊕ 𝑈) = 𝑈)
Theorem	lsmub1 19453	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))
Theorem	lsmub2 19454	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))
Theorem	lsmunss 19455	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)) → (𝑇 ∪ 𝑈) ⊆ (𝑇 ⊕ 𝑈))
Theorem	lsmless1 19456	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) ∧ 𝑆 ⊆ 𝑇) → (𝑆 ⊕ 𝑈) ⊆ (𝑇 ⊕ 𝑈))
Theorem	lsmless2 19457	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) ∧ 𝑇 ⊆ 𝑈) → (𝑆 ⊕ 𝑇) ⊆ (𝑆 ⊕ 𝑈))
Theorem	lsmless12 19458	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)) ∧ (𝑅 ⊆ 𝑆 ∧ 𝑇 ⊆ 𝑈)) → (𝑅 ⊕ 𝑇) ⊆ (𝑆 ⊕ 𝑈))
Theorem	lsmidm 19459	Subgroup sum is idempotent. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) (Proof shortened by AV, 27-Dec-2023.)
⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ (𝑈 ∈ (SubGrp‘𝐺) → (𝑈 ⊕ 𝑈) = 𝑈)
Theorem	lsmlub 19460	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)) → ((𝑆 ⊆ 𝑈 ∧ 𝑇 ⊆ 𝑈) ↔ (𝑆 ⊕ 𝑇) ⊆ 𝑈))
Theorem	lsmss1 19461	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) ∧ 𝑇 ⊆ 𝑈) → (𝑇 ⊕ 𝑈) = 𝑈)
Theorem	lsmss1b 19462	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)) → (𝑇 ⊆ 𝑈 ↔ (𝑇 ⊕ 𝑈) = 𝑈))
Theorem	lsmss2 19463	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) ∧ 𝑈 ⊆ 𝑇) → (𝑇 ⊕ 𝑈) = 𝑇)
Theorem	lsmss2b 19464	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)) → (𝑈 ⊆ 𝑇 ↔ (𝑇 ⊕ 𝑈) = 𝑇))
Theorem	lsmass 19465	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)))
Theorem	mndlsmidm 19466	Subgroup sum is idempotent for monoids. This corresponds to the observation in [Lang] p. 6. (Contributed by AV, 27-Dec-2023.)
⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐺 ∈ Mnd → (𝐵 ⊕ 𝐵) = 𝐵)
Theorem	lsm01 19467	Subgroup sum with the zero subgroup. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ 0 = (0g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ (𝑋 ∈ (SubGrp‘𝐺) → (𝑋 ⊕ { 0 }) = 𝑋)
Theorem	lsm02 19468	Subgroup sum with the zero subgroup. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ 0 = (0g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ (𝑋 ∈ (SubGrp‘𝐺) → ({ 0 } ⊕ 𝑋) = 𝑋)
Theorem	subglsm 19469	The subgroup sum evaluated within a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝐻 = (𝐺 ↾s 𝑆)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 𝐴 = (LSSum‘𝐻)    ⇒   ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ 𝑆 ∧ 𝑈 ⊆ 𝑆) → (𝑇 ⊕ 𝑈) = (𝑇𝐴𝑈))
Theorem	lssnle 19470	Equivalent expressions for "not less than". (chnlei 30490 analog.) (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    ⇒   ⊢ (𝜑 → (¬ 𝑈 ⊆ 𝑇 ↔ 𝑇 ⊊ (𝑇 ⊕ 𝑈)))
Theorem	lsmmod 19471	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)) ∧ 𝑆 ⊆ 𝑈) → (𝑆 ⊕ (𝑇 ∩ 𝑈)) = ((𝑆 ⊕ 𝑇) ∩ 𝑈))
Theorem	lsmmod2 19472	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)) ∧ 𝑈 ⊆ 𝑆) → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = ((𝑆 ∩ 𝑇) ⊕ 𝑈))
Theorem	lsmpropd 19473*	If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 29-Jun-2015.) (Revised by AV, 25-Apr-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐿 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (LSSum‘𝐾) = (LSSum‘𝐿))
Theorem	cntzrecd 19474	Commute the "subgroups commute" predicate. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    ⇒   ⊢ (𝜑 → 𝑈 ⊆ (𝑍‘𝑇))
Theorem	lsmcntz 19475	The "subgroups commute" predicate applied to a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ 𝑍 = (Cntz‘𝐺)    ⇒   ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ⊆ (𝑍‘𝑈) ↔ (𝑆 ⊆ (𝑍‘𝑈) ∧ 𝑇 ⊆ (𝑍‘𝑈))))
Theorem	lsmcntzr 19476	The "subgroups commute" predicate applied to a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ 𝑍 = (Cntz‘𝐺)    ⇒   ⊢ (𝜑 → (𝑆 ⊆ (𝑍‘(𝑇 ⊕ 𝑈)) ↔ (𝑆 ⊆ (𝑍‘𝑇) ∧ 𝑆 ⊆ (𝑍‘𝑈))))
Theorem	lsmdisj 19477	Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 })    ⇒   ⊢ (𝜑 → ((𝑆 ∩ 𝑈) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 }))
Theorem	lsmdisj2 19478	Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → (𝑆 ∩ 𝑇) = { 0 })    ⇒   ⊢ (𝜑 → (𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 })
Theorem	lsmdisj3 19479	Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → (𝑆 ∩ 𝑇) = { 0 })    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑆 ⊆ (𝑍‘𝑇))    ⇒   ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 })
Theorem	lsmdisjr 19480	Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 })    ⇒   ⊢ (𝜑 → ((𝑆 ∩ 𝑇) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 }))
Theorem	lsmdisj2r 19481	Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 })    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    ⇒   ⊢ (𝜑 → ((𝑆 ⊕ 𝑈) ∩ 𝑇) = { 0 })
Theorem	lsmdisj3r 19482	Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 })    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    ⇒   ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 })
Theorem	lsmdisj2a 19483	Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝜑 → ((((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 }) ↔ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })))
Theorem	lsmdisj2b 19484	Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝜑 → ((((𝑆 ⊕ 𝑈) ∩ 𝑇) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 }) ↔ ((𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 })))
Theorem	lsmdisj3a 19485	Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑆 ⊆ (𝑍‘𝑇))    ⇒   ⊢ (𝜑 → ((((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 }) ↔ ((𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 })))
Theorem	lsmdisj3b 19486	Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    ⇒   ⊢ (𝜑 → ((((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 }) ↔ ((𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 })))
Theorem	subgdisj1 19487	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.)
⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    &   ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐶)
Theorem	subgdisj2 19488	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.)
⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    &   ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷))    ⇒   ⊢ (𝜑 → 𝐵 = 𝐷)
Theorem	subgdisjb 19489	Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. Analogous to opth 5438, this theorem shows a way of representing a pair of vectors. (Contributed by NM, 5-Jul-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
Theorem	pj1fval 19490*	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)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 𝑃 = (proj1‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))))
Theorem	pj1val 19491*	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)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 𝑃 = (proj1‘𝐺)    ⇒   ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) = (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)))
Theorem	pj1eu 19492*	Uniqueness of a left projection. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ∃!𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦))
Theorem	pj1f 19493	The left projection function maps a direct subspace sum onto the left factor. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ 𝑃 = (proj1‘𝐺)    ⇒   ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
Theorem	pj2f 19494	The right projection function maps a direct subspace sum onto the right factor. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ 𝑃 = (proj1‘𝐺)    ⇒   ⊢ (𝜑 → (𝑈𝑃𝑇):(𝑇 ⊕ 𝑈)⟶𝑈)
Theorem	pj1id 19495	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.)
⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ 𝑃 = (proj1‘𝐺)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))
Theorem	pj1eq 19496	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.)
⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ 𝑃 = (proj1‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑇 ⊕ 𝑈))    &   ⊢ (𝜑 → 𝐵 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑋 = (𝐵 + 𝐶) ↔ (((𝑇𝑃𝑈)‘𝑋) = 𝐵 ∧ ((𝑈𝑃𝑇)‘𝑋) = 𝐶)))
Theorem	pj1lid 19497	The left projection function is the identity on the left subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ 𝑃 = (proj1‘𝐺)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑇) → ((𝑇𝑃𝑈)‘𝑋) = 𝑋)
Theorem	pj1rid 19498	The left projection function is the zero operator on the right subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ 𝑃 = (proj1‘𝐺)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → ((𝑇𝑃𝑈)‘𝑋) = 0 )
Theorem	pj1ghm 19499	The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ 𝑃 = (proj1‘𝐺)    ⇒   ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺 ↾s (𝑇 ⊕ 𝑈)) GrpHom 𝐺))
Theorem	pj1ghm2 19500	The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ 𝑃 = (proj1‘𝐺)    ⇒   ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺 ↾s (𝑇 ⊕ 𝑈)) GrpHom (𝐺 ↾s 𝑇)))