tendoi - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > tendoi		Structured version Visualization version GIF version

Theorem tendoi 40397

Description: Value of inverse endomorphism. (Contributed by NM, 12-Jun-2013.)

**Hypotheses**
Ref	Expression
tendoi.i	⊢ 𝐼 = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ^◡(𝑠‘𝑓)))
tendoi.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
tendoi	⊢ (𝑆 ∈ 𝐸 → (𝐼‘𝑆) = (𝑔 ∈ 𝑇 ↦ ^◡(𝑆‘𝑔)))

Distinct variable groups: 𝐸,𝑠 𝑓,𝑔,𝑠,𝑇 𝑓,𝑊,𝑔,𝑠 𝑆,𝑔 Allowed substitution hints: 𝑆(𝑓,𝑠) 𝐸(𝑓,𝑔) 𝐼(𝑓,𝑔,𝑠) 𝐾(𝑓,𝑔,𝑠)

Proof of Theorem tendoi Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6895	. . . 4 ⊢ (𝑢 = 𝑆 → (𝑢‘𝑔) = (𝑆‘𝑔))
2	1	cnveqd 5878	. . 3 ⊢ (𝑢 = 𝑆 → ^◡(𝑢‘𝑔) = ^◡(𝑆‘𝑔))
3	2	mpteq2dv 5251	. 2 ⊢ (𝑢 = 𝑆 → (𝑔 ∈ 𝑇 ↦ ^◡(𝑢‘𝑔)) = (𝑔 ∈ 𝑇 ↦ ^◡(𝑆‘𝑔)))
4		tendoi.i	. . 3 ⊢ 𝐼 = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ^◡(𝑠‘𝑓)))
5	4	tendoicbv 40396	. 2 ⊢ 𝐼 = (𝑢 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ^◡(𝑢‘𝑔)))
6		tendoi.t	. 2 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
7	3, 5, 6	mptfvmpt 7240	1 ⊢ (𝑆 ∈ 𝐸 → (𝐼‘𝑆) = (𝑔 ∈ 𝑇 ↦ ^◡(𝑆‘𝑔)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ↦ cmpt 5232 ^◡ccnv 5677 ‘cfv 6549 LTrncltrn 39704

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-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5429

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-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557

This theorem is referenced by: tendoi2 40398 tendoicl 40399

	
		OSZAR »