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Theorem List for Metamath Proof Explorer - 18201-18300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremefgredlemg 18201* Lemma for efgred 18207. (Contributed by Mario Carneiro, 4-Jun-2016.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))    &   (𝜑𝐴 ∈ dom 𝑆)    &   (𝜑𝐵 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐴) = (𝑆𝐵))    &   (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   𝐾 = (((#‘𝐴) − 1) − 1)    &   𝐿 = (((#‘𝐵) − 1) − 1)    &   (𝜑𝑃 ∈ (0...(#‘(𝐴𝐾))))    &   (𝜑𝑄 ∈ (0...(#‘(𝐵𝐿))))    &   (𝜑𝑈 ∈ (𝐼 × 2𝑜))    &   (𝜑𝑉 ∈ (𝐼 × 2𝑜))    &   (𝜑 → (𝑆𝐴) = (𝑃(𝑇‘(𝐴𝐾))𝑈))    &   (𝜑 → (𝑆𝐵) = (𝑄(𝑇‘(𝐵𝐿))𝑉))       (𝜑 → (#‘(𝐴𝐾)) = (#‘(𝐵𝐿)))

Theoremefgredleme 18202* Lemma for efgred 18207. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))    &   (𝜑𝐴 ∈ dom 𝑆)    &   (𝜑𝐵 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐴) = (𝑆𝐵))    &   (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   𝐾 = (((#‘𝐴) − 1) − 1)    &   𝐿 = (((#‘𝐵) − 1) − 1)    &   (𝜑𝑃 ∈ (0...(#‘(𝐴𝐾))))    &   (𝜑𝑄 ∈ (0...(#‘(𝐵𝐿))))    &   (𝜑𝑈 ∈ (𝐼 × 2𝑜))    &   (𝜑𝑉 ∈ (𝐼 × 2𝑜))    &   (𝜑 → (𝑆𝐴) = (𝑃(𝑇‘(𝐴𝐾))𝑈))    &   (𝜑 → (𝑆𝐵) = (𝑄(𝑇‘(𝐵𝐿))𝑉))    &   (𝜑 → ¬ (𝐴𝐾) = (𝐵𝐿))    &   (𝜑𝑃 ∈ (ℤ‘(𝑄 + 2)))    &   (𝜑𝐶 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐶) = (((𝐵𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴𝐾))⟩)))       (𝜑 → ((𝐴𝐾) ∈ ran (𝑇‘(𝑆𝐶)) ∧ (𝐵𝐿) ∈ ran (𝑇‘(𝑆𝐶))))

Theoremefgredlemd 18203* The reduced word that forms the base of the sequence in efgsval 18190 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))    &   (𝜑𝐴 ∈ dom 𝑆)    &   (𝜑𝐵 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐴) = (𝑆𝐵))    &   (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   𝐾 = (((#‘𝐴) − 1) − 1)    &   𝐿 = (((#‘𝐵) − 1) − 1)    &   (𝜑𝑃 ∈ (0...(#‘(𝐴𝐾))))    &   (𝜑𝑄 ∈ (0...(#‘(𝐵𝐿))))    &   (𝜑𝑈 ∈ (𝐼 × 2𝑜))    &   (𝜑𝑉 ∈ (𝐼 × 2𝑜))    &   (𝜑 → (𝑆𝐴) = (𝑃(𝑇‘(𝐴𝐾))𝑈))    &   (𝜑 → (𝑆𝐵) = (𝑄(𝑇‘(𝐵𝐿))𝑉))    &   (𝜑 → ¬ (𝐴𝐾) = (𝐵𝐿))    &   (𝜑𝑃 ∈ (ℤ‘(𝑄 + 2)))    &   (𝜑𝐶 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐶) = (((𝐵𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴𝐾))⟩)))       (𝜑 → (𝐴‘0) = (𝐵‘0))

Theoremefgredlemc 18204* The reduced word that forms the base of the sequence in efgsval 18190 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))    &   (𝜑𝐴 ∈ dom 𝑆)    &   (𝜑𝐵 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐴) = (𝑆𝐵))    &   (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   𝐾 = (((#‘𝐴) − 1) − 1)    &   𝐿 = (((#‘𝐵) − 1) − 1)    &   (𝜑𝑃 ∈ (0...(#‘(𝐴𝐾))))    &   (𝜑𝑄 ∈ (0...(#‘(𝐵𝐿))))    &   (𝜑𝑈 ∈ (𝐼 × 2𝑜))    &   (𝜑𝑉 ∈ (𝐼 × 2𝑜))    &   (𝜑 → (𝑆𝐴) = (𝑃(𝑇‘(𝐴𝐾))𝑈))    &   (𝜑 → (𝑆𝐵) = (𝑄(𝑇‘(𝐵𝐿))𝑉))    &   (𝜑 → ¬ (𝐴𝐾) = (𝐵𝐿))       (𝜑 → (𝑃 ∈ (ℤ𝑄) → (𝐴‘0) = (𝐵‘0)))

Theoremefgredlemb 18205* The reduced word that forms the base of the sequence in efgsval 18190 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))    &   (𝜑𝐴 ∈ dom 𝑆)    &   (𝜑𝐵 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐴) = (𝑆𝐵))    &   (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   𝐾 = (((#‘𝐴) − 1) − 1)    &   𝐿 = (((#‘𝐵) − 1) − 1)    &   (𝜑𝑃 ∈ (0...(#‘(𝐴𝐾))))    &   (𝜑𝑄 ∈ (0...(#‘(𝐵𝐿))))    &   (𝜑𝑈 ∈ (𝐼 × 2𝑜))    &   (𝜑𝑉 ∈ (𝐼 × 2𝑜))    &   (𝜑 → (𝑆𝐴) = (𝑃(𝑇‘(𝐴𝐾))𝑈))    &   (𝜑 → (𝑆𝐵) = (𝑄(𝑇‘(𝐵𝐿))𝑉))    &   (𝜑 → ¬ (𝐴𝐾) = (𝐵𝐿))        ¬ 𝜑

Theoremefgredlem 18206* The reduced word that forms the base of the sequence in efgsval 18190 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))    &   (𝜑𝐴 ∈ dom 𝑆)    &   (𝜑𝐵 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐴) = (𝑆𝐵))    &   (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))        ¬ 𝜑

Theoremefgred 18207* The reduced word that forms the base of the sequence in efgsval 18190 is uniquely determined, given the terminal point. (Contributed by Mario Carneiro, 28-Sep-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       ((𝐴 ∈ dom 𝑆𝐵 ∈ dom 𝑆 ∧ (𝑆𝐴) = (𝑆𝐵)) → (𝐴‘0) = (𝐵‘0))

Theoremefgrelexlema 18208* If two words 𝐴, 𝐵 are related under the free group equivalence, then there exist two extension sequences 𝑎, 𝑏 such that 𝑎 ends at 𝐴, 𝑏 ends at 𝐵, and 𝑎 and 𝐵 have the same starting point. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   𝐿 = {⟨𝑖, 𝑗⟩ ∣ ∃𝑐 ∈ (𝑆 “ {𝑖})∃𝑑 ∈ (𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0)}       (𝐴𝐿𝐵 ↔ ∃𝑎 ∈ (𝑆 “ {𝐴})∃𝑏 ∈ (𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0))

Theoremefgrelexlemb 18209* If two words 𝐴, 𝐵 are related under the free group equivalence, then there exist two extension sequences 𝑎, 𝑏 such that 𝑎 ends at 𝐴, 𝑏 ends at 𝐵, and 𝑎 and 𝐵 have the same starting point. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   𝐿 = {⟨𝑖, 𝑗⟩ ∣ ∃𝑐 ∈ (𝑆 “ {𝑖})∃𝑑 ∈ (𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0)}        𝐿

Theoremefgrelex 18210* If two words 𝐴, 𝐵 are related under the free group equivalence, then there exist two extension sequences 𝑎, 𝑏 such that 𝑎 ends at 𝐴, 𝑏 ends at 𝐵, and 𝑎 and 𝐵 have the same starting point. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       (𝐴 𝐵 → ∃𝑎 ∈ (𝑆 “ {𝐴})∃𝑏 ∈ (𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0))

Theoremefgredeu 18211* There is a unique reduced word equivalent to a given word. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       (𝐴𝑊 → ∃!𝑑𝐷 𝑑 𝐴)

Theoremefgred2 18212* Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       ((𝐴 ∈ dom 𝑆𝐵 ∈ dom 𝑆) → ((𝑆𝐴) (𝑆𝐵) ↔ (𝐴‘0) = (𝐵‘0)))

Theoremefgcpbllema 18213* Lemma for efgrelex 18210. Define an auxiliary equivalence relation 𝐿 such that 𝐴𝐿𝐵 if there are sequences from 𝐴 to 𝐵 passing through the same reduced word. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   𝐿 = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑊 ∧ ((𝐴 ++ 𝑖) ++ 𝐵) ((𝐴 ++ 𝑗) ++ 𝐵))}       (𝑋𝐿𝑌 ↔ (𝑋𝑊𝑌𝑊 ∧ ((𝐴 ++ 𝑋) ++ 𝐵) ((𝐴 ++ 𝑌) ++ 𝐵)))

Theoremefgcpbllemb 18214* Lemma for efgrelex 18210. Show that 𝐿 is an equivalence relation containing all direct extensions of a word, so is closed under . (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   𝐿 = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑊 ∧ ((𝐴 ++ 𝑖) ++ 𝐵) ((𝐴 ++ 𝑗) ++ 𝐵))}       ((𝐴𝑊𝐵𝑊) → 𝐿)

Theoremefgcpbl 18215* Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       ((𝐴𝑊𝐵𝑊𝑋 𝑌) → ((𝐴 ++ 𝑋) ++ 𝐵) ((𝐴 ++ 𝑌) ++ 𝐵))

Theoremefgcpbl2 18216* Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       ((𝐴 𝑋𝐵 𝑌) → (𝐴 ++ 𝐵) (𝑋 ++ 𝑌))

Theoremfrgpval 18217 Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝐺 = (freeGrp‘𝐼)    &   𝑀 = (freeMnd‘(𝐼 × 2𝑜))    &    = ( ~FG𝐼)       (𝐼𝑉𝐺 = (𝑀 /s ))

Theoremfrgpcpbl 18218 Compatibility of the group operation with the free group equivalence relation. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝐺 = (freeGrp‘𝐼)    &   𝑀 = (freeMnd‘(𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &    + = (+g𝑀)       ((𝐴 𝐶𝐵 𝐷) → (𝐴 + 𝐵) (𝐶 + 𝐷))

Theoremfrgp0 18219 The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝐺 = (freeGrp‘𝐼)    &    = ( ~FG𝐼)       (𝐼𝑉 → (𝐺 ∈ Grp ∧ [∅] = (0g𝐺)))

Theoremfrgpeccl 18220 Closure of the quotient map in a free group. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝐺 = (freeGrp‘𝐼)    &    = ( ~FG𝐼)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &   𝐵 = (Base‘𝐺)       (𝑋𝑊 → [𝑋] 𝐵)

Theoremfrgpgrp 18221 The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝐺 = (freeGrp‘𝐼)       (𝐼𝑉𝐺 ∈ Grp)

Theoremfrgpadd 18222 Addition in the free group is given by concatenation. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &   𝐺 = (freeGrp‘𝐼)    &    = ( ~FG𝐼)    &    + = (+g𝐺)       ((𝐴𝑊𝐵𝑊) → ([𝐴] + [𝐵] ) = [(𝐴 ++ 𝐵)] )

Theoremfrgpinv 18223* The inverse of an element of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &   𝐺 = (freeGrp‘𝐼)    &    = ( ~FG𝐼)    &   𝑁 = (invg𝐺)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)       (𝐴𝑊 → (𝑁‘[𝐴] ) = [(𝑀 ∘ (reverse‘𝐴))] )

Theoremfrgpmhm 18224* The "natural map" from words of the free monoid to their cosets in the free group is a surjective monoid homomorphism. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑀 = (freeMnd‘(𝐼 × 2𝑜))    &   𝑊 = (Base‘𝑀)    &   𝐺 = (freeGrp‘𝐼)    &    = ( ~FG𝐼)    &   𝐹 = (𝑥𝑊 ↦ [𝑥] )       (𝐼𝑉𝐹 ∈ (𝑀 MndHom 𝐺))

Theoremvrgpfval 18225* The canonical injection from the generating set 𝐼 to the base set of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
= ( ~FG𝐼)    &   𝑈 = (varFGrp𝐼)       (𝐼𝑉𝑈 = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))

Theoremvrgpval 18226 The value of the generating elements of a free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
= ( ~FG𝐼)    &   𝑈 = (varFGrp𝐼)       ((𝐼𝑉𝐴𝐼) → (𝑈𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )

Theoremvrgpf 18227 The mapping from the index set to the generators is a function into the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
= ( ~FG𝐼)    &   𝑈 = (varFGrp𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑋 = (Base‘𝐺)       (𝐼𝑉𝑈:𝐼𝑋)

Theoremvrgpinv 18228 The inverse of a generating element is represented by 𝐴, 1⟩ instead of 𝐴, 0⟩. (Contributed by Mario Carneiro, 2-Oct-2015.)
= ( ~FG𝐼)    &   𝑈 = (varFGrp𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑁 = (invg𝐺)       ((𝐼𝑉𝐴𝐼) → (𝑁‘(𝑈𝐴)) = [⟨“⟨𝐴, 1𝑜⟩”⟩] )

Theoremfrgpuptf 18229* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)       (𝜑𝑇:(𝐼 × 2𝑜)⟶𝐵)

Theoremfrgpuptinv 18230* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)       ((𝜑𝐴 ∈ (𝐼 × 2𝑜)) → (𝑇‘(𝑀𝐴)) = (𝑁‘(𝑇𝐴)))

Theoremfrgpuplem 18231* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)       ((𝜑𝐴 𝐶) → (𝐻 Σg (𝑇𝐴)) = (𝐻 Σg (𝑇𝐶)))

Theoremfrgpupf 18232* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑋 = (Base‘𝐺)    &   𝐸 = ran (𝑔𝑊 ↦ ⟨[𝑔] , (𝐻 Σg (𝑇𝑔))⟩)       (𝜑𝐸:𝑋𝐵)

Theoremfrgpupval 18233* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑋 = (Base‘𝐺)    &   𝐸 = ran (𝑔𝑊 ↦ ⟨[𝑔] , (𝐻 Σg (𝑇𝑔))⟩)       ((𝜑𝐴𝑊) → (𝐸‘[𝐴] ) = (𝐻 Σg (𝑇𝐴)))

Theoremfrgpup1 18234* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑋 = (Base‘𝐺)    &   𝐸 = ran (𝑔𝑊 ↦ ⟨[𝑔] , (𝐻 Σg (𝑇𝑔))⟩)       (𝜑𝐸 ∈ (𝐺 GrpHom 𝐻))

Theoremfrgpup2 18235* The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑋 = (Base‘𝐺)    &   𝐸 = ran (𝑔𝑊 ↦ ⟨[𝑔] , (𝐻 Σg (𝑇𝑔))⟩)    &   𝑈 = (varFGrp𝐼)    &   (𝜑𝐴𝐼)       (𝜑 → (𝐸‘(𝑈𝐴)) = (𝐹𝐴))

Theoremfrgpup3lem 18236* The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑋 = (Base‘𝐺)    &   𝐸 = ran (𝑔𝑊 ↦ ⟨[𝑔] , (𝐻 Σg (𝑇𝑔))⟩)    &   𝑈 = (varFGrp𝐼)    &   (𝜑𝐾 ∈ (𝐺 GrpHom 𝐻))    &   (𝜑 → (𝐾𝑈) = 𝐹)       (𝜑𝐾 = 𝐸)

Theoremfrgpup3 18237* Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝐺 = (freeGrp‘𝐼)    &   𝐵 = (Base‘𝐻)    &   𝑈 = (varFGrp𝐼)       ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → ∃!𝑚 ∈ (𝐺 GrpHom 𝐻)(𝑚𝑈) = 𝐹)

Theorem0frgp 18238 The free group on zero generators is trivial. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐺 = (freeGrp‘∅)    &   𝐵 = (Base‘𝐺)       𝐵 ≈ 1𝑜

10.3  Abelian groups

10.3.1  Definition and basic properties

Syntaxccmn 18239 Extend class notation with class of all commutative monoids.
class CMnd

Syntaxcabl 18240 Extend class notation with class of all Abelian groups.
class Abel

Definitiondf-cmn 18241* Define class of all commutative monoids. (Contributed by Mario Carneiro, 6-Jan-2015.)
CMnd = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)(𝑎(+g𝑔)𝑏) = (𝑏(+g𝑔)𝑎)}

Definitiondf-abl 18242 Define class of all Abelian groups. (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
Abel = (Grp ∩ CMnd)

Theoremisabl 18243 The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.)
(𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))

Theoremablgrp 18244 An Abelian group is a group. (Contributed by NM, 26-Aug-2011.)
(𝐺 ∈ Abel → 𝐺 ∈ Grp)

Theoremablcmn 18245 An Abelian group is a commutative monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
(𝐺 ∈ Abel → 𝐺 ∈ CMnd)

Theoremiscmn 18246* The predicate "is a commutative monoid." (Contributed by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))

Theoremisabl2 18247* The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))

Theoremcmnpropd 18248* If two structures have the same group components (properties), one is a commutative monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))       (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd))

Theoremablpropd 18249* If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 6-Dec-2014.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))       (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))

Theoremablprop 18250 If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 11-Oct-2013.)
(Base‘𝐾) = (Base‘𝐿)    &   (+g𝐾) = (+g𝐿)       (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)

Theoremiscmnd 18251* Properties that determine a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑+ = (+g𝐺))    &   (𝜑𝐺 ∈ Mnd)    &   ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))       (𝜑𝐺 ∈ CMnd)

Theoremisabld 18252* Properties that determine an Abelian group. (Contributed by NM, 6-Aug-2013.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑+ = (+g𝐺))    &   (𝜑𝐺 ∈ Grp)    &   ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))       (𝜑𝐺 ∈ Abel)

Theoremisabli 18253* Properties that determine an Abelian group. (Contributed by NM, 4-Sep-2011.)
𝐺 ∈ Grp    &   𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))       𝐺 ∈ Abel

Theoremcmnmnd 18254 A commutative monoid is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
(𝐺 ∈ CMnd → 𝐺 ∈ Mnd)

Theoremcmncom 18255 A commutative monoid is commutative. (Contributed by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ CMnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Theoremablcom 18256 An Abelian group operation is commutative. (Contributed by NM, 26-Aug-2011.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Theoremcmn32 18257 Commutative/associative law for Abelian groups. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))

Theoremcmn4 18258 Commutative/associative law for Abelian groups. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))

Theoremcmn12 18259 Commutative/associative law for Abelian monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))

Theoremabl32 18260 Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))

Theoremablinvadd 18261 The inverse of an Abelian group operation. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑁 = (invg𝐺)       ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → (𝑁‘(𝑋 + 𝑌)) = ((𝑁𝑋) + (𝑁𝑌)))

Theoremablsub2inv 18262 Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   𝑁 = (invg𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑁𝑋) (𝑁𝑌)) = (𝑌 𝑋))

Theoremablsubadd 18263 Relationship between Abelian group subtraction and addition. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) = 𝑍 ↔ (𝑌 + 𝑍) = 𝑋))

Theoremablsub4 18264 Commutative/associative subtraction law for Abelian groups. (Contributed by NM, 31-Mar-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) (𝑍 + 𝑊)) = ((𝑋 𝑍) + (𝑌 𝑊)))

𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 + 𝑌) = (𝑍 + 𝑊) ↔ (𝑋 𝑍) = (𝑊 𝑌)))

Theoremabladdsub 18266 Associative-type law for group subtraction and addition. (Contributed by NM, 19-Apr-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) 𝑍) = ((𝑋 𝑍) + 𝑌))

Theoremablpncan2 18267 Cancellation law for subtraction. (Contributed by NM, 2-Oct-2014.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 + 𝑌) 𝑋) = 𝑌)

Theoremablpncan3 18268 A cancellation law for commutative groups. (Contributed by NM, 23-Mar-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 + (𝑌 𝑋)) = 𝑌)

Theoremablsubsub 18269 Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) + 𝑍))

Theoremablsubsub4 18270 Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 + 𝑍)))

Theoremablpnpcan 18271 Cancellation law for mixed addition and subtraction. (pnpcan 10358 analog.) (Contributed by NM, 29-May-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 + 𝑌) (𝑋 + 𝑍)) = (𝑌 𝑍))

Theoremablnncan 18272 Cancellation law for group subtraction. (nncan 10348 analog.) (Contributed by NM, 7-Apr-2015.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 (𝑋 𝑌)) = 𝑌)

Theoremablsub32 18273 Swap the second and third terms in a double group subtraction. (Contributed by NM, 7-Apr-2015.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 𝑌) 𝑍) = ((𝑋 𝑍) 𝑌))

Theoremablnnncan 18274 Cancellation law for group subtraction. (nnncan 10354 analog.) (Contributed by NM, 29-Feb-2008.) (Revised by AV, 27-Aug-2021.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 (𝑌 𝑍)) 𝑍) = (𝑋 𝑌))

Theoremablnnncan1 18275 Cancellation law for group subtraction. (nnncan1 10355 analog.) (Contributed by NM, 7-Apr-2015.)
𝐵 = (Base‘𝐺)    &    = (-g𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 𝑌) (𝑋 𝑍)) = (𝑍 𝑌))

Theoremablsubsub23 18276 Swap subtrahend and result of group subtraction. (Contributed by NM, 14-Dec-2007.) (Revised by AV, 7-Oct-2021.)
𝑉 = (Base‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) = 𝐶 ↔ (𝐴 𝐶) = 𝐵))

Theoremmulgnn0di 18277 Group multiple of a sum, for nonnegative multiples. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ CMnd ∧ (𝑀 ∈ ℕ0𝑋𝐵𝑌𝐵)) → (𝑀 · (𝑋 + 𝑌)) = ((𝑀 · 𝑋) + (𝑀 · 𝑌)))

Theoremmulgdi 18278 Group multiple of a sum. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Abel ∧ (𝑀 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑀 · (𝑋 + 𝑌)) = ((𝑀 · 𝑋) + (𝑀 · 𝑌)))

Theoremmulgmhm 18279* The map from 𝑥 to 𝑛𝑥 for a fixed positive integer 𝑛 is a monoid homomorphism if the monoid is commutative. (Contributed by Mario Carneiro, 4-May-2015.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ CMnd ∧ 𝑀 ∈ ℕ0) → (𝑥𝐵 ↦ (𝑀 · 𝑥)) ∈ (𝐺 MndHom 𝐺))

Theoremmulgghm 18280* The map from 𝑥 to 𝑛𝑥 for a fixed integer 𝑛 is a group homomorphism if the group is commutative. (Contributed by Mario Carneiro, 4-May-2015.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)       ((𝐺 ∈ Abel ∧ 𝑀 ∈ ℤ) → (𝑥𝐵 ↦ (𝑀 · 𝑥)) ∈ (𝐺 GrpHom 𝐺))

Theoremmulgsubdi 18281 Group multiple of a difference. (Contributed by Mario Carneiro, 13-Dec-2014.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &    = (-g𝐺)       ((𝐺 ∈ Abel ∧ (𝑀 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑀 · (𝑋 𝑌)) = ((𝑀 · 𝑋) (𝑀 · 𝑌)))

Theoremghmfghm 18282* The function fulfilling the conditions of ghmgrp 17586 is a group homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020.)
𝑋 = (Base‘𝐺)    &   𝑌 = (Base‘𝐻)    &    + = (+g𝐺)    &    = (+g𝐻)    &   ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝐺 ∈ Grp)       (𝜑𝐹 ∈ (𝐺 GrpHom 𝐻))

Theoremghmcmn 18283* The image of a commutative monoid 𝐺 under a group homomorphism 𝐹 is a commutative monoid. (Contributed by Thierry Arnoux, 26-Jan-2020.)
𝑋 = (Base‘𝐺)    &   𝑌 = (Base‘𝐻)    &    + = (+g𝐺)    &    = (+g𝐻)    &   ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝐺 ∈ CMnd)       (𝜑𝐻 ∈ CMnd)

Theoremghmabl 18284* The image of an abelian group 𝐺 under a group homomorphism 𝐹 is an abelian group. (Contributed by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.)
𝑋 = (Base‘𝐺)    &   𝑌 = (Base‘𝐻)    &    + = (+g𝐺)    &    = (+g𝐻)    &   ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &   (𝜑𝐹:𝑋onto𝑌)    &   (𝜑𝐺 ∈ Abel)       (𝜑𝐻 ∈ Abel)

Theoreminvghm 18285 The inversion map is a group automorphism if and only if the group is abelian. (In general it is only a group homomorphism into the opposite group, but in an abelian group the opposite group coincides with the group itself.) (Contributed by Mario Carneiro, 4-May-2015.)
𝐵 = (Base‘𝐺)    &   𝐼 = (invg𝐺)       (𝐺 ∈ Abel ↔ 𝐼 ∈ (𝐺 GrpHom 𝐺))

Theoremeqgabl 18286 Value of the subgroup coset equivalence relation on an abelian group. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑋 = (Base‘𝐺)    &    = (-g𝐺)    &    = (𝐺 ~QG 𝑆)       ((𝐺 ∈ Abel ∧ 𝑆𝑋) → (𝐴 𝐵 ↔ (𝐴𝑋𝐵𝑋 ∧ (𝐵 𝐴) ∈ 𝑆)))

Theoremsubgabl 18287 A subgroup of an abelian group is also abelian. (Contributed by Mario Carneiro, 3-Dec-2014.)
𝐻 = (𝐺s 𝑆)       ((𝐺 ∈ Abel ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Abel)

Theoremsubcmn 18288 A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝐻 = (𝐺s 𝑆)       ((𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd) → 𝐻 ∈ CMnd)

Theoremsubmcmn 18289 A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝐻 = (𝐺s 𝑆)       ((𝐺 ∈ CMnd ∧ 𝑆 ∈ (SubMnd‘𝐺)) → 𝐻 ∈ CMnd)

Theoremsubmcmn2 18290 A submonoid is commutative iff it is a subset of its own centralizer. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝐻 = (𝐺s 𝑆)    &   𝑍 = (Cntz‘𝐺)       (𝑆 ∈ (SubMnd‘𝐺) → (𝐻 ∈ CMnd ↔ 𝑆 ⊆ (𝑍𝑆)))

Theoremcntzcmn 18291 The centralizer of any subset in a commutative monoid is the whole monoid. (Contributed by Mario Carneiro, 3-Oct-2015.)
𝐵 = (Base‘𝐺)    &   𝑍 = (Cntz‘𝐺)       ((𝐺 ∈ CMnd ∧ 𝑆𝐵) → (𝑍𝑆) = 𝐵)

Theoremcntzcmnss 18292 Any subset in a commutative monoid is a subset of its centralizer. (Contributed by AV, 12-Jan-2019.)
𝐵 = (Base‘𝐺)    &   𝑍 = (Cntz‘𝐺)       ((𝐺 ∈ CMnd ∧ 𝑆𝐵) → 𝑆 ⊆ (𝑍𝑆))

Theoremcntzspan 18293 If the generators commute, the generated monoid is commutative. (Contributed by Mario Carneiro, 25-Apr-2016.)
𝑍 = (Cntz‘𝐺)    &   𝐾 = (mrCls‘(SubMnd‘𝐺))    &   𝐻 = (𝐺s (𝐾𝑆))       ((𝐺 ∈ Mnd ∧ 𝑆 ⊆ (𝑍𝑆)) → 𝐻 ∈ CMnd)

Theoremcntzcmnf 18294 Discharge the centralizer assumption in a commutative monoid. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝐵 = (Base‘𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))

Theoremghmplusg 18295 The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
+ = (+g𝑁)       ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹𝑓 + 𝐺) ∈ (𝑀 GrpHom 𝑁))

Theoremablnsg 18296 Every subgroup of an abelian group is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
(𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺))

Theoremodadd1 18297 The order of a product in an abelian group divides the LCM of the orders of the factors. (Contributed by Mario Carneiro, 20-Oct-2015.)
𝑂 = (od‘𝐺)    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂𝐴) gcd (𝑂𝐵))) ∥ ((𝑂𝐴) · (𝑂𝐵)))

Theoremodadd2 18298 The order of a product in an abelian group is divisible by the LCM of the orders of the factors divided by the GCD. (Contributed by Mario Carneiro, 20-Oct-2015.)
𝑂 = (od‘𝐺)    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) → ((𝑂𝐴) · (𝑂𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂𝐴) gcd (𝑂𝐵))↑2)))

Theoremodadd 18299 The order of a product is the product of the orders, if the factors have coprime order. (Contributed by Mario Carneiro, 20-Oct-2015.)
𝑂 = (od‘𝐺)    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)       (((𝐺 ∈ Abel ∧ 𝐴𝑋𝐵𝑋) ∧ ((𝑂𝐴) gcd (𝑂𝐵)) = 1) → (𝑂‘(𝐴 + 𝐵)) = ((𝑂𝐴) · (𝑂𝐵)))

Theoremgex2abl 18300 A group with exponent 2 (or 1) is abelian. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑋 = (Base‘𝐺)    &   𝐸 = (gEx‘𝐺)       ((𝐺 ∈ Grp ∧ 𝐸 ∥ 2) → 𝐺 ∈ Abel)

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