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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendoi2 | Structured version Visualization version GIF version |
Description: Value of additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.) |
Ref | Expression |
---|---|
tendoi.i | ⊢ 𝐼 = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓))) |
tendoi.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
tendoi2 | ⊢ ((𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((𝐼‘𝑆)‘𝐹) = ◡(𝑆‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tendoi.i | . . . 4 ⊢ 𝐼 = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓))) | |
2 | tendoi.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
3 | 1, 2 | tendoi 40177 | . . 3 ⊢ (𝑆 ∈ 𝐸 → (𝐼‘𝑆) = (𝑔 ∈ 𝑇 ↦ ◡(𝑆‘𝑔))) |
4 | 3 | adantr 480 | . 2 ⊢ ((𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝐼‘𝑆) = (𝑔 ∈ 𝑇 ↦ ◡(𝑆‘𝑔))) |
5 | fveq2 6884 | . . . 4 ⊢ (𝑔 = 𝐹 → (𝑆‘𝑔) = (𝑆‘𝐹)) | |
6 | 5 | cnveqd 5868 | . . 3 ⊢ (𝑔 = 𝐹 → ◡(𝑆‘𝑔) = ◡(𝑆‘𝐹)) |
7 | 6 | adantl 481 | . 2 ⊢ (((𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 = 𝐹) → ◡(𝑆‘𝑔) = ◡(𝑆‘𝐹)) |
8 | simpr 484 | . 2 ⊢ ((𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ 𝑇) | |
9 | fvex 6897 | . . . 4 ⊢ (𝑆‘𝐹) ∈ V | |
10 | 9 | cnvex 7912 | . . 3 ⊢ ◡(𝑆‘𝐹) ∈ V |
11 | 10 | a1i 11 | . 2 ⊢ ((𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ◡(𝑆‘𝐹) ∈ V) |
12 | 4, 7, 8, 11 | fvmptd 6998 | 1 ⊢ ((𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((𝐼‘𝑆)‘𝐹) = ◡(𝑆‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ↦ cmpt 5224 ◡ccnv 5668 ‘cfv 6536 LTrncltrn 39484 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 |
This theorem is referenced by: tendoicl 40179 tendoipl 40180 dihjatcclem4 40804 |
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