P. 328 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32701-32800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	wrdsplex 32701*	Existence of a split of a word at a given index. (Contributed by Thierry Arnoux, 11-Oct-2018.) (Proof shortened by AV, 3-Nov-2022.)
⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ∃𝑣 ∈ Word 𝑆𝑊 = ((𝑊 ↾ (0..^𝑁)) ++ 𝑣))
Theorem	pfx1s2 32702	The prefix of length 1 of a length 2 word. (Contributed by Thierry Arnoux, 19-Sep-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (⟨“𝐴𝐵”⟩ prefix 1) = ⟨“𝐴”⟩)
Theorem	pfxrn2 32703	The range of a prefix of a word is a subset of the range of that word. Stronger version of pfxrn 14662. (Contributed by Thierry Arnoux, 12-Dec-2023.)
⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) ⊆ ran 𝑊)
Theorem	pfxrn3 32704	Express the range of a prefix of a word. Stronger version of pfxrn2 32703. (Contributed by Thierry Arnoux, 13-Dec-2023.)
⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) = (𝑊 “ (0..^𝐿)))
Theorem	pfxf1 32705	Condition for a prefix to be injective. (Contributed by Thierry Arnoux, 13-Dec-2023.)
⊢ (𝜑 → 𝑊 ∈ Word 𝑆)    &   ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝑆)    &   ⊢ (𝜑 → 𝐿 ∈ (0...(♯‘𝑊)))    ⇒   ⊢ (𝜑 → (𝑊 prefix 𝐿):dom (𝑊 prefix 𝐿)–1-1→𝑆)
Theorem	s1f1 32706	Conditions for a length 1 string to be a one-to-one function. (Contributed by Thierry Arnoux, 11-Dec-2023.)
⊢ (𝜑 → 𝐼 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ⟨“𝐼”⟩:dom ⟨“𝐼”⟩–1-1→𝐷)
Theorem	s2rn 32707	Range of a length 2 string. (Contributed by Thierry Arnoux, 19-Sep-2023.)
⊢ (𝜑 → 𝐼 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ran ⟨“𝐼𝐽”⟩ = {𝐼, 𝐽})
Theorem	s2f1 32708	Conditions for a length 2 string to be a one-to-one function. (Contributed by Thierry Arnoux, 19-Sep-2023.)
⊢ (𝜑 → 𝐼 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐼 ≠ 𝐽)    ⇒   ⊢ (𝜑 → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1→𝐷)
Theorem	s3rn 32709	Range of a length 3 string. (Contributed by Thierry Arnoux, 19-Sep-2023.)
⊢ (𝜑 → 𝐼 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ran ⟨“𝐼𝐽𝐾”⟩ = {𝐼, 𝐽, 𝐾})
Theorem	s3f1 32710	Conditions for a length 3 string to be a one-to-one function. (Contributed by Thierry Arnoux, 19-Sep-2023.)
⊢ (𝜑 → 𝐼 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐼 ≠ 𝐽)    &   ⊢ (𝜑 → 𝐽 ≠ 𝐾)    &   ⊢ (𝜑 → 𝐾 ≠ 𝐼)    ⇒   ⊢ (𝜑 → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩–1-1→𝐷)
Theorem	s3clhash 32711	Closure of the words of length 3 in a preimage using the hash function. (Contributed by Thierry Arnoux, 27-Sep-2023.)
⊢ ⟨“𝐼𝐽𝐾”⟩ ∈ (◡♯ “ {3})
Theorem	ccatf1 32712	Conditions for a concatenation to be injective. (Contributed by Thierry Arnoux, 11-Dec-2023.)
⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ Word 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ Word 𝑆)    &   ⊢ (𝜑 → 𝐴:dom 𝐴–1-1→𝑆)    &   ⊢ (𝜑 → 𝐵:dom 𝐵–1-1→𝑆)    &   ⊢ (𝜑 → (ran 𝐴 ∩ ran 𝐵) = ∅)    ⇒   ⊢ (𝜑 → (𝐴 ++ 𝐵):dom (𝐴 ++ 𝐵)–1-1→𝑆)
Theorem	pfxlsw2ccat 32713	Reconstruct a word from its prefix and its last two symbols. (Contributed by Thierry Arnoux, 26-Sep-2023.)
⊢ 𝑁 = (♯‘𝑊)    ⇒   ⊢ ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ 𝑁) → 𝑊 = ((𝑊 prefix (𝑁 − 2)) ++ ⟨“(𝑊‘(𝑁 − 2))(𝑊‘(𝑁 − 1))”⟩))
Theorem	wrdt2ind 32714*	Perform an induction over the structure of a word of even length. (Contributed by Thierry Arnoux, 26-Sep-2023.)
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 ++ ⟨“𝑖𝑗”⟩) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ ((𝑦 ∈ Word 𝐵 ∧ 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵) → (𝜒 → 𝜃))    ⇒   ⊢ ((𝐴 ∈ Word 𝐵 ∧ 2 ∥ (♯‘𝐴)) → 𝜏)
Theorem	swrdrn2 32715	The range of a subword is a subset of the range of that word. Stronger version of swrdrn 14629. (Contributed by Thierry Arnoux, 12-Dec-2023.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝑀, 𝑁⟩) ⊆ ran 𝑊)
Theorem	swrdrn3 32716	Express the range of a subword. Stronger version of swrdrn2 32715. (Contributed by Thierry Arnoux, 13-Dec-2023.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑊 “ (𝑀..^𝑁)))
Theorem	swrdf1 32717	Condition for a subword to be injective. (Contributed by Thierry Arnoux, 12-Dec-2023.)
⊢ (𝜑 → 𝑊 ∈ Word 𝐷)    &   ⊢ (𝜑 → 𝑀 ∈ (0...𝑁))    &   ⊢ (𝜑 → 𝑁 ∈ (0...(♯‘𝑊)))    &   ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)    ⇒   ⊢ (𝜑 → (𝑊 substr ⟨𝑀, 𝑁⟩):dom (𝑊 substr ⟨𝑀, 𝑁⟩)–1-1→𝐷)
Theorem	swrdrndisj 32718	Condition for the range of two subwords of an injective word to be disjoint. (Contributed by Thierry Arnoux, 13-Dec-2023.)
⊢ (𝜑 → 𝑊 ∈ Word 𝐷)    &   ⊢ (𝜑 → 𝑀 ∈ (0...𝑁))    &   ⊢ (𝜑 → 𝑁 ∈ (0...(♯‘𝑊)))    &   ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)    &   ⊢ (𝜑 → 𝑂 ∈ (𝑁...𝑃))    &   ⊢ (𝜑 → 𝑃 ∈ (𝑁...(♯‘𝑊)))    ⇒   ⊢ (𝜑 → (ran (𝑊 substr ⟨𝑀, 𝑁⟩) ∩ ran (𝑊 substr ⟨𝑂, 𝑃⟩)) = ∅)
21.3.7.1  Splicing words (substring replacement)
Theorem	splfv3 32719	Symbols to the right of a splice are unaffected. (Contributed by Thierry Arnoux, 14-Dec-2023.)
⊢ (𝜑 → 𝑆 ∈ Word 𝐴)    &   ⊢ (𝜑 → 𝐹 ∈ (0...𝑇))    &   ⊢ (𝜑 → 𝑇 ∈ (0...(♯‘𝑆)))    &   ⊢ (𝜑 → 𝑅 ∈ Word 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ (0..^((♯‘𝑆) − 𝑇)))    &   ⊢ (𝜑 → 𝐾 = (𝐹 + (♯‘𝑅)))    ⇒   ⊢ (𝜑 → ((𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩)‘(𝑋 + 𝐾)) = (𝑆‘(𝑋 + 𝑇)))
21.3.7.2  Cyclic shift of words
Theorem	1cshid 32720	Cyclically shifting a single letter word keeps it unchanged. (Contributed by Thierry Arnoux, 21-Nov-2023.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ ∧ (♯‘𝑊) = 1) → (𝑊 cyclShift 𝑁) = 𝑊)
Theorem	cshw1s2 32721	Cyclically shifting a length 2 word swaps its symbols. (Contributed by Thierry Arnoux, 19-Sep-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (⟨“𝐴𝐵”⟩ cyclShift 1) = ⟨“𝐵𝐴”⟩)
Theorem	cshwrnid 32722	Cyclically shifting a word preserves its range. (Contributed by Thierry Arnoux, 19-Sep-2023.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℤ) → ran (𝑊 cyclShift 𝑁) = ran 𝑊)
Theorem	cshf1o 32723	Condition for the cyclic shift to be a bijection. (Contributed by Thierry Arnoux, 4-Oct-2023.)
⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷 ∧ 𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁):dom 𝑊–1-1-onto→ran 𝑊)
21.3.8  Extensible Structures
21.3.8.1  Structure restriction operator
Theorem	ressplusf 32724	The group operation function +𝑓 of a structure's restriction is the operation function's restriction to the new base. (Contributed by Thierry Arnoux, 26-Mar-2017.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ ⨣ = (+g‘𝐺)    &   ⊢ ⨣ Fn (𝐵 × 𝐵)    &   ⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ (+𝑓‘𝐻) = ( ⨣ ↾ (𝐴 × 𝐴))
Theorem	ressnm 32725	The norm in a restricted structure. (Contributed by Thierry Arnoux, 8-Oct-2017.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑁 = (norm‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝑁 ↾ 𝐴) = (norm‘𝐻))
Theorem	abvpropd2 32726	Weaker version of abvpropd 20721. (Contributed by Thierry Arnoux, 8-Nov-2017.)
⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   ⊢ (𝜑 → (+g‘𝐾) = (+g‘𝐿))    &   ⊢ (𝜑 → (.r‘𝐾) = (.r‘𝐿))    ⇒   ⊢ (𝜑 → (AbsVal‘𝐾) = (AbsVal‘𝐿))
21.3.8.2  The opposite group
Theorem	oppgle 32727	less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ ≤ = (le‘𝑅)    ⇒   ⊢ ≤ = (le‘𝑂)
Theorem	oppgleOLD 32728	Obsolete version of oppgle 32727 as of 27-Oct-2024. less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ ≤ = (le‘𝑅)    ⇒   ⊢ ≤ = (le‘𝑂)
Theorem	oppglt 32729	less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ < = (lt‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → < = (lt‘𝑂))
21.3.8.3  Posets
Theorem	ressprs 32730	The restriction of a proset is a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.)
⊢ 𝐵 = (Base‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝐾 ↾s 𝐴) ∈ Proset )
Theorem	oduprs 32731	Being a proset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.)
⊢ 𝐷 = (ODual‘𝐾)    ⇒   ⊢ (𝐾 ∈ Proset → 𝐷 ∈ Proset )
Theorem	posrasymb 32732	A poset ordering is asymetric. (Contributed by Thierry Arnoux, 13-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    ⇒   ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌))
Theorem	resspos 32733	The restriction of a Poset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
⊢ ((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) → (𝐹 ↾s 𝐴) ∈ Poset)
Theorem	resstos 32734	The restriction of a Toset is a Toset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
⊢ ((𝐹 ∈ Toset ∧ 𝐴 ∈ 𝑉) → (𝐹 ↾s 𝐴) ∈ Toset)
Theorem	odutos 32735	Being a toset is a self-dual property. (Contributed by Thierry Arnoux, 13-Sep-2018.)
⊢ 𝐷 = (ODual‘𝐾)    ⇒   ⊢ (𝐾 ∈ Toset → 𝐷 ∈ Toset)
Theorem	tlt2 32736	In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 13-Apr-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ∨ 𝑌 < 𝑋))
Theorem	tlt3 32737	In a Toset, two elements must compare strictly, or be equal. (Contributed by Thierry Arnoux, 13-Apr-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 = 𝑌 ∨ 𝑋 < 𝑌 ∨ 𝑌 < 𝑋))
Theorem	trleile 32738	In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 12-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))    ⇒   ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋))
Theorem	toslublem 32739*	Lemma for toslub 32740 and xrsclat 32778. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by NM, 15-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ Toset)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑏 ≤ 𝑐 → 𝑎 ≤ 𝑐)) ↔ (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏 < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏 < 𝑑))))
Theorem	toslub 32740	In a toset, the lowest upper bound lub, defined for partial orders is the supremum, sup(𝐴, 𝐵, < ), defined for total orders. (these are the set.mm definitions: lowest upper bound and supremum are normally synonymous). Note that those two values are also equal if such a supremum does not exist: in that case, both are equal to the empty set. (Contributed by Thierry Arnoux, 15-Feb-2018.) (Revised by Thierry Arnoux, 24-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ Toset)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ((lub‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < ))
Theorem	tosglblem 32741*	Lemma for tosglb 32742 and xrsclat 32778. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by NM, 15-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ Toset)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((∀𝑏 ∈ 𝐴 𝑎 ≤ 𝑏 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑐 ≤ 𝑏 → 𝑐 ≤ 𝑎)) ↔ (∀𝑏 ∈ 𝐴 ¬ 𝑎◡ < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏◡ < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏◡ < 𝑑))))
Theorem	tosglb 32742	Same theorem as toslub 32740, for infinimum. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by AV, 28-Sep-2020.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ Toset)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ((glb‘𝐾)‘𝐴) = inf(𝐴, 𝐵, < ))
21.3.8.4  Complete lattices
Theorem	clatp0cl 32743	The poset zero of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0.‘𝑊)    ⇒   ⊢ (𝑊 ∈ CLat → 0 ∈ 𝐵)
Theorem	clatp1cl 32744	The poset one of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 1 = (1.‘𝑊)    ⇒   ⊢ (𝑊 ∈ CLat → 1 ∈ 𝐵)
21.3.8.5  Order Theory
Syntax	cmnt 32745	Extend class notation with monotone functions.
class Monot
Syntax	cmgc 32746	Extend class notation with the monotone Galois connection.
class MGalConn
Definition	df-mnt 32747*	Define a monotone function between two ordered sets. (Contributed by Thierry Arnoux, 20-Apr-2024.)
⊢ Monot = (𝑣 ∈ V, 𝑤 ∈ V ↦ ⦋(Base‘𝑣) / 𝑎⦌{𝑓 ∈ ((Base‘𝑤) ↑m 𝑎) ∣ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓‘𝑥)(le‘𝑤)(𝑓‘𝑦))})
Definition	df-mgc 32748*	Define monotone Galois connections. See mgcval 32754 for an expanded version. (Contributed by Thierry Arnoux, 20-Apr-2024.)
⊢ MGalConn = (𝑣 ∈ V, 𝑤 ∈ V ↦ ⦋(Base‘𝑣) / 𝑎⦌⦋(Base‘𝑤) / 𝑏⦌{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏 ↑m 𝑎) ∧ 𝑔 ∈ (𝑎 ↑m 𝑏)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑓‘𝑥)(le‘𝑤)𝑦 ↔ 𝑥(le‘𝑣)(𝑔‘𝑦)))})
Theorem	mntoval 32749*	Operation value of the monotone function. (Contributed by Thierry Arnoux, 23-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉Monot𝑊) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))})
Theorem	ismnt 32750*	Express the statement "𝐹 is monotone". (Contributed by Thierry Arnoux, 23-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))))
Theorem	ismntd 32751	Property of being a monotone increasing function, deduction version. (Contributed by Thierry Arnoux, 24-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ 𝐶)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑉Monot𝑊))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    ⇒   ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑌))
Theorem	mntf 32752	A monotone function is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ∧ 𝐹 ∈ (𝑉Monot𝑊)) → 𝐹:𝐴⟶𝐵)
Theorem	mgcoval 32753*	Operation value of the monotone Galois connection. (Contributed by Thierry Arnoux, 23-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉MGalConn𝑊) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))})
Theorem	mgcval 32754*	Monotone Galois connection between two functions 𝐹 and 𝐺. If this relation is satisfied, 𝐹 is called the lower adjoint of 𝐺, and 𝐺 is called the upper adjoint of 𝐹. Technically, this is implemented as an operation taking a pair of structures 𝑉 and 𝑊, expected to be posets, which gives a relation between pairs of functions 𝐹 and 𝐺. If such a relation exists, it can be proven to be unique. Galois connections generalize the fundamental theorem of Galois theory about the correspondence between subgroups and subfields. (Contributed by Thierry Arnoux, 23-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    ⇒   ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))))
Theorem	mgcf1 32755	The lower adjoint 𝐹 of a Galois connection is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    ⇒   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
Theorem	mgcf2 32756	The upper adjoint 𝐺 of a Galois connection is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    ⇒   ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
Theorem	mgccole1 32757	An inequality for the kernel operator 𝐺 ∘ 𝐹. (Contributed by Thierry Arnoux, 26-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝑋 ≤ (𝐺‘(𝐹‘𝑋)))
Theorem	mgccole2 32758	Inequality for the closure operator (𝐹 ∘ 𝐺) of the Galois connection 𝐻. (Contributed by Thierry Arnoux, 26-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹‘(𝐺‘𝑌)) ≲ 𝑌)
Theorem	mgcmnt1 32759	The lower adjoint 𝐹 of a Galois connection is monotonically increasing. (Contributed by Thierry Arnoux, 26-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    ⇒   ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑌))
Theorem	mgcmnt2 32760	The upper adjoint 𝐺 of a Galois connection is monotonically increasing. (Contributed by Thierry Arnoux, 26-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≲ 𝑌)    ⇒   ⊢ (𝜑 → (𝐺‘𝑋) ≤ (𝐺‘𝑌))
Theorem	mgcmntco 32761*	A Galois connection like statement, for two functions with same range. (Contributed by Thierry Arnoux, 26-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    &   ⊢ 𝐶 = (Base‘𝑋)    &   ⊢ < = (le‘𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ Proset )    &   ⊢ (𝜑 → 𝐾 ∈ (𝑉Monot𝑋))    &   ⊢ (𝜑 → 𝐿 ∈ (𝑊Monot𝑋))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥)) ↔ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)))
Theorem	dfmgc2lem 32762*	Lemma for dfmgc2, backwards direction. (Contributed by Thierry Arnoux, 26-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))    &   ⊢ (𝜑 → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))    &   ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐵) → (𝐹‘(𝐺‘𝑢)) ≲ 𝑢)    ⇒   ⊢ (𝜑 → 𝐹𝐻𝐺)
Theorem	dfmgc2 32763*	Alternate definition of the monotone Galois connection. (Contributed by Thierry Arnoux, 26-Apr-2024.)
⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    ⇒   ⊢ (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣))) ∧ (∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))))))
Theorem	mgcmnt1d 32764	Galois connection implies monotonicity of the left adjoint. (Contributed by Thierry Arnoux, 21-Jul-2024.)
⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑉Monot𝑊))
Theorem	mgcmnt2d 32765	Galois connection implies monotonicity of the right adjoint. (Contributed by Thierry Arnoux, 21-Jul-2024.)
⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Proset )    &   ⊢ (𝜑 → 𝑊 ∈ Proset )    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝑊Monot𝑉))
Theorem	mgccnv 32766	The inverse Galois connection is the Galois connection of the dual orders. (Contributed by Thierry Arnoux, 26-Apr-2024.)
⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ 𝑀 = ((ODual‘𝑊)MGalConn(ODual‘𝑉))    ⇒   ⊢ ((𝑉 ∈ Proset ∧ 𝑊 ∈ Proset ) → (𝐹𝐻𝐺 ↔ 𝐺𝑀𝐹))
Theorem	pwrssmgc 32767*	Given a function 𝐹, exhibit a Galois connection between subsets of its domain and subsets of its range. (Contributed by Thierry Arnoux, 26-Apr-2024.)
⊢ 𝐺 = (𝑛 ∈ 𝒫 𝑌 ↦ (◡𝐹 “ 𝑛))    &   ⊢ 𝐻 = (𝑚 ∈ 𝒫 𝑋 ↦ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑚})    &   ⊢ 𝑉 = (toInc‘𝒫 𝑌)    &   ⊢ 𝑊 = (toInc‘𝒫 𝑋)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)    ⇒   ⊢ (𝜑 → 𝐺(𝑉MGalConn𝑊)𝐻)
Theorem	mgcf1olem1 32768	Property of a Galois connection, lemma for mgcf1o 32770. (Contributed by Thierry Arnoux, 26-Jul-2024.)
⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Poset)    &   ⊢ (𝜑 → 𝑊 ∈ Poset)    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) = (𝐹‘𝑋))
Theorem	mgcf1olem2 32769	Property of a Galois connection, lemma for mgcf1o 32770. (Contributed by Thierry Arnoux, 26-Jul-2024.)
⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Poset)    &   ⊢ (𝜑 → 𝑊 ∈ Poset)    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺‘(𝐹‘(𝐺‘𝑌))) = (𝐺‘𝑌))
Theorem	mgcf1o 32770	Given a Galois connection, exhibit an order isomorphism. (Contributed by Thierry Arnoux, 26-Jul-2024.)
⊢ 𝐻 = (𝑉MGalConn𝑊)    &   ⊢ 𝐴 = (Base‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ ≤ = (le‘𝑉)    &   ⊢ ≲ = (le‘𝑊)    &   ⊢ (𝜑 → 𝑉 ∈ Poset)    &   ⊢ (𝜑 → 𝑊 ∈ Poset)    &   ⊢ (𝜑 → 𝐹𝐻𝐺)    ⇒   ⊢ (𝜑 → (𝐹 ↾ ran 𝐺) Isom ≤ , ≲ (ran 𝐺, ran 𝐹))
21.3.8.6  Extended reals Structure - misc additions
Axiom	ax-xrssca 32771	Assume the scalar component of the extended real structure is the field of the real numbers (this has to be defined in the main body of set.mm). (Contributed by Thierry Arnoux, 22-Oct-2017.)
⊢ ℝfld = (Scalar‘ℝ*𝑠)
Axiom	ax-xrsvsca 32772	Assume the scalar product of the extended real structure is the extended real number multiplication operation (this has to be defined in the main body of set.mm). (Contributed by Thierry Arnoux, 22-Oct-2017.)
⊢ ·e = ( ·𝑠 ‘ℝ*𝑠)
Theorem	xrs0 32773	The zero of the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 13255 and df-xrs 17478), however it has a zero. (Contributed by Thierry Arnoux, 13-Jun-2017.)
⊢ 0 = (0g‘ℝ*𝑠)
Theorem	xrslt 32774	The "strictly less than" relation for the extended real structure. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ < = (lt‘ℝ*𝑠)
Theorem	xrsinvgval 32775	The inversion operation in the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 13255 and df-xrs 17478), however it has an inversion operation. (Contributed by Thierry Arnoux, 13-Jun-2017.)
⊢ (𝐵 ∈ ℝ* → ((invg‘ℝ*𝑠)‘𝐵) = -𝑒𝐵)
Theorem	xrsmulgzz 32776	The "multiple" function in the extended real numbers structure. (Contributed by Thierry Arnoux, 14-Jun-2017.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ*) → (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵))
Theorem	xrstos 32777	The extended real numbers form a toset. (Contributed by Thierry Arnoux, 15-Feb-2018.)
⊢ ℝ*𝑠 ∈ Toset
Theorem	xrsclat 32778	The extended real numbers form a complete lattice. (Contributed by Thierry Arnoux, 15-Feb-2018.)
⊢ ℝ*𝑠 ∈ CLat
Theorem	xrsp0 32779	The poset 0 of the extended real numbers is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.) (Proof shortened by AV, 28-Sep-2020.)
⊢ -∞ = (0.‘ℝ*𝑠)
Theorem	xrsp1 32780	The poset 1 of the extended real numbers is plus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)
⊢ +∞ = (1.‘ℝ*𝑠)
21.3.8.7  The extended nonnegative real numbers commutative monoid
Theorem	xrge0base 32781	The base of the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 30-Jan-2017.)
⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞)))
Theorem	xrge00 32782	The zero of the extended nonnegative real numbers monoid. (Contributed by Thierry Arnoux, 30-Jan-2017.)
⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞)))
Theorem	xrge0plusg 32783	The additive law of the extended nonnegative real numbers monoid is the addition in the extended real numbers. (Contributed by Thierry Arnoux, 20-Mar-2017.)
⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞)))
Theorem	xrge0le 32784	The "less than or equal to" relation in the extended real numbers. (Contributed by Thierry Arnoux, 14-Mar-2018.)
⊢ ≤ = (le‘(ℝ*𝑠 ↾s (0[,]+∞)))
Theorem	xrge0mulgnn0 32785	The group multiple function in the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 14-Jun-2017.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ (0[,]+∞)) → (𝐴(.g‘(ℝ*𝑠 ↾s (0[,]+∞)))𝐵) = (𝐴 ·e 𝐵))
Theorem	xrge0addass 32786	Associativity of extended nonnegative real addition. (Contributed by Thierry Arnoux, 8-Jun-2017.)
⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → ((𝐴 +𝑒 𝐵) +𝑒 𝐶) = (𝐴 +𝑒 (𝐵 +𝑒 𝐶)))
Theorem	xrge0addgt0 32787	The sum of nonnegative and positive numbers is positive. See addgtge0 11727. (Contributed by Thierry Arnoux, 6-Jul-2017.)
⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞)) ∧ 0 < 𝐴) → 0 < (𝐴 +𝑒 𝐵))
Theorem	xrge0adddir 32788	Right-distributivity of extended nonnegative real multiplication over addition. (Contributed by Thierry Arnoux, 30-Jun-2017.)
⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → ((𝐴 +𝑒 𝐵) ·e 𝐶) = ((𝐴 ·e 𝐶) +𝑒 (𝐵 ·e 𝐶)))
Theorem	xrge0adddi 32789	Left-distributivity of extended nonnegative real multiplication over addition. (Contributed by Thierry Arnoux, 6-Sep-2018.)
⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → (𝐶 ·e (𝐴 +𝑒 𝐵)) = ((𝐶 ·e 𝐴) +𝑒 (𝐶 ·e 𝐵)))
Theorem	xrge0npcan 32790	Extended nonnegative real version of npcan 11494. (Contributed by Thierry Arnoux, 9-Jun-2017.)
⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴)
Theorem	fsumrp0cl 32791*	Closure of a finite sum of nonnegative reals. (Contributed by Thierry Arnoux, 25-Jun-2017.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞))
21.3.9  Algebra
21.3.9.1  Monoids
Theorem	cmn4d 32792	Commutative/associative law for commutative monoids. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))
Theorem	cmn246135 32793	Rearrange terms in a commutative monoid sum. Lemma for rlocaddval 33005. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑉 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 + 𝑌) + ((𝑍 + 𝑈) + (𝑉 + 𝑊))) = ((𝑌 + (𝑈 + 𝑊)) + (𝑋 + (𝑍 + 𝑉))))
Theorem	cmn145236 32794	Rearrange terms in a commutative monoid sum. Lemma for rlocaddval 33005. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑉 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 + 𝑌) + ((𝑍 + 𝑈) + (𝑉 + 𝑊))) = ((𝑋 + (𝑈 + 𝑉)) + (𝑌 + (𝑍 + 𝑊))))
Theorem	submcld 32795	Submonoids are closed under the monoid operation. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ + = (+g‘𝑀)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝑀))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑆)
21.3.9.2  Monoids Homomorphisms
Theorem	abliso 32796	The image of an Abelian group by a group isomorphism is also Abelian. (Contributed by Thierry Arnoux, 8-Mar-2018.)
⊢ ((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Abel)
Theorem	lmhmghmd 32797	A module homomorphism is a group homomorphism. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇))
Theorem	mhmimasplusg 32798	Value of the operation of the surjective image. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝑊 = (𝐹 “s 𝑉)    &   ⊢ 𝐵 = (Base‘𝑉)    &   ⊢ 𝐶 = (Base‘𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹:𝐵–onto→𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑉 MndHom 𝑊))    &   ⊢ + = (+g‘𝑉)    &   ⊢ ⨣ = (+g‘𝑊)    ⇒   ⊢ (𝜑 → ((𝐹‘𝑋) ⨣ (𝐹‘𝑌)) = (𝐹‘(𝑋 + 𝑌)))
Theorem	lmhmimasvsca 32799	Value of the scalar product of the surjective image of a module. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝑊 = (𝐹 “s 𝑉)    &   ⊢ 𝐵 = (Base‘𝑉)    &   ⊢ 𝐶 = (Base‘𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹:𝐵–onto→𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑉 LMHom 𝑊))    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ × = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘(Scalar‘𝑉))    ⇒   ⊢ (𝜑 → (𝑋 × (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌)))
21.3.9.3  Finitely supported group sums - misc additions
Theorem	gsumsubg 32800	The group sum in a subgroup is the same as the group sum. (Contributed by Thierry Arnoux, 28-May-2023.)
⊢ 𝐻 = (𝐺 ↾s 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘𝐺))    ⇒   ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))