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Mirrors > Home > ILE Home > Th. List > shftidt2 | GIF version |
Description: Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.) |
Ref | Expression |
---|---|
shftfval.1 | ⊢ 𝐹 ∈ V |
Ref | Expression |
---|---|
shftidt2 | ⊢ (𝐹 shift 0) = (𝐹 ↾ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subid1 7982 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥 − 0) = 𝑥) | |
2 | 1 | breq1d 3939 | . . . 4 ⊢ (𝑥 ∈ ℂ → ((𝑥 − 0)𝐹𝑦 ↔ 𝑥𝐹𝑦)) |
3 | 2 | pm5.32i 449 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ (𝑥 − 0)𝐹𝑦) ↔ (𝑥 ∈ ℂ ∧ 𝑥𝐹𝑦)) |
4 | 3 | opabbii 3995 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 0)𝐹𝑦)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑥𝐹𝑦)} |
5 | 0cn 7758 | . . 3 ⊢ 0 ∈ ℂ | |
6 | shftfval.1 | . . . 4 ⊢ 𝐹 ∈ V | |
7 | 6 | shftfval 10593 | . . 3 ⊢ (0 ∈ ℂ → (𝐹 shift 0) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 0)𝐹𝑦)}) |
8 | 5, 7 | ax-mp 5 | . 2 ⊢ (𝐹 shift 0) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 0)𝐹𝑦)} |
9 | dfres2 4871 | . 2 ⊢ (𝐹 ↾ ℂ) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑥𝐹𝑦)} | |
10 | 4, 8, 9 | 3eqtr4i 2170 | 1 ⊢ (𝐹 shift 0) = (𝐹 ↾ ℂ) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∈ wcel 1480 Vcvv 2686 class class class wbr 3929 {copab 3988 ↾ cres 4541 (class class class)co 5774 ℂcc 7618 0cc0 7620 − cmin 7933 shift cshi 10586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-resscn 7712 ax-1cn 7713 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7935 df-shft 10587 |
This theorem is referenced by: shftidt 10605 |
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