P. 199 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 19801-19900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pgphash 19801	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 19802*	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 19803	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 19804	A Sylow 𝑃-subgroup is a subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))
Theorem	slwispgp 19805	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 19806	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 19807	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 19808*	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 19809	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 19810	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 19811*	Lemma for sylow2a 19813. 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 19812*	Lemma for sylow2a 19813. 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 19813*	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 19814*	Lemma for sylow2b 19817. 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 19815*	Lemma for sylow2b 19817. 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 19816*	Sylow's second theorem. Putting together the results of sylow2a 19813 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 19817*	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 19818	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 19819	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 19820*	Sylow's second theorem. See also sylow2b 19817 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 19819). 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 19821*	Lemma for sylow3 19827, 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 19822*	Lemma for sylow3 19827, 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 19823*	Lemma for sylow3 19827, 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 19824*	Lemma for sylow3 19827, 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 19825*	Lemma for sylow3 19827, second part. Reduce the group action of sylow3lem1 19821 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 19826*	Lemma for sylow3 19827, second part. Using the lemma sylow2a 19813, 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 19827	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 19828	Extend class notation with subgroup sum.
class LSSum
Syntax	cpj1 19829	Extend class notation with left projection.
class proj1
Definition	df-lsm 19830*	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 19831*	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 19832*	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 19833*	Subspace sum value (for a group or vector space). Extended domain version of lsmval 19842. (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 19834*	Subspace sum membership (for a group or vector space). Extended domain version of lsmelval 19843. (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 19835	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 19836	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 19837	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 19838	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 19839	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 19840	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 19841	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 19842*	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 19843*	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 19844	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 19845*	Subgroup sum membership analogue of lsmelval 19843 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 19846	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 19847	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 19848	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 19849	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 19850	The direct product is idempotent for submonoids. (Contributed by AV, 27-Dec-2023.)
⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ (𝑈 ∈ (SubMnd‘𝐺) → (𝑈 ⊕ 𝑈) = 𝑈)
Theorem	lsmub1 19851	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 19852	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 19853	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 19854	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 19855	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 19856	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 19857	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 19858	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 19859	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 19860	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 19861	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 19862	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 19863	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 19864	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 19865	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 19866	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 19867	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 19868	Equivalent expressions for "not less than". (chnlei 32069 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 19869	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 19870	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 19871*	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 19872	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 19873	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 19874	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 19875	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 19876	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 19877	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 19878	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 19879	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 19880	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 19881	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 19882	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 19883	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 19884	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 19885	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 19886	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 19887	Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. Analogous to opth 5445, 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 19888*	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 19889*	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 19890*	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 19891	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 19892	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 19893	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 19894	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 19895	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 19896	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 19897	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 19898	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)))
Theorem	lsmhash 19899	The order of the direct product of groups. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ ⊕ = (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 → 𝑇 ∈ Fin)    &   ⊢ (𝜑 → 𝑈 ∈ Fin)    ⇒   ⊢ (𝜑 → (♯‘(𝑇 ⊕ 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))
10.2.13  Free groups
Syntax	cefg 19900	Extend class notation with the free group equivalence relation.
class ~FG