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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 7328 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥 − 0) = 𝑥) | |
2 | 1 | breq1d 3795 | . . . 4 ⊢ (𝑥 ∈ ℂ → ((𝑥 − 0)𝐹𝑦 ↔ 𝑥𝐹𝑦)) |
3 | 2 | pm5.32i 441 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ (𝑥 − 0)𝐹𝑦) ↔ (𝑥 ∈ ℂ ∧ 𝑥𝐹𝑦)) |
4 | 3 | opabbii 3845 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 0)𝐹𝑦)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑥𝐹𝑦)} |
5 | 0cn 7111 | . . 3 ⊢ 0 ∈ ℂ | |
6 | shftfval.1 | . . . 4 ⊢ 𝐹 ∈ V | |
7 | 6 | shftfval 9709 | . . 3 ⊢ (0 ∈ ℂ → (𝐹 shift 0) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 0)𝐹𝑦)}) |
8 | 5, 7 | ax-mp 7 | . 2 ⊢ (𝐹 shift 0) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 0)𝐹𝑦)} |
9 | dfres2 4678 | . 2 ⊢ (𝐹 ↾ ℂ) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑥𝐹𝑦)} | |
10 | 4, 8, 9 | 3eqtr4i 2111 | 1 ⊢ (𝐹 shift 0) = (𝐹 ↾ ℂ) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 = wceq 1284 ∈ wcel 1433 Vcvv 2601 class class class wbr 3785 {copab 3838 ↾ cres 4365 (class class class)co 5532 ℂcc 6979 0cc0 6981 − cmin 7279 shift cshi 9702 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-resscn 7068 ax-1cn 7069 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-sub 7281 df-shft 9703 |
This theorem is referenced by: shftidt 9721 |
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