negslem1 - Mathbox for Richard Penner

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Theorem negslem1 42854

Description: An equivalence between identically restricted order-reversing self-isometries. (Contributed by RP, 30-Sep-2024.)

**Assertion**
Ref	Expression
negslem1	⊢ (𝐴 = 𝐵 → ((𝐹 ↾ 𝐴) Isom 𝑅, ^◡𝑅(𝐴, 𝐴) ↔ (𝐹 ↾ 𝐵) Isom 𝑅, ^◡𝑅(𝐵, 𝐵)))

**Proof of Theorem negslem1**
Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
2	1, 1	resisoeq45d 42853	1 ⊢ (𝐴 = 𝐵 → ((𝐹 ↾ 𝐴) Isom 𝑅, ^◡𝑅(𝐴, 𝐴) ↔ (𝐹 ↾ 𝐵) Isom 𝑅, ^◡𝑅(𝐵, 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ^◡ccnv 5679 ↾ cres 5682 Isom wiso 6552

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2698

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560

This theorem is referenced by: (None)

	
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