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Mirrors > Home > MPE Home > Th. List > shftidt2 | Structured version Visualization version 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 10180 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥 − 0) = 𝑥) | |
2 | 1 | breq1d 4593 | . . . 4 ⊢ (𝑥 ∈ ℂ → ((𝑥 − 0)𝐹𝑦 ↔ 𝑥𝐹𝑦)) |
3 | 2 | pm5.32i 667 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ (𝑥 − 0)𝐹𝑦) ↔ (𝑥 ∈ ℂ ∧ 𝑥𝐹𝑦)) |
4 | 3 | opabbii 4649 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 0)𝐹𝑦)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑥𝐹𝑦)} |
5 | 0cn 9911 | . . 3 ⊢ 0 ∈ ℂ | |
6 | shftfval.1 | . . . 4 ⊢ 𝐹 ∈ V | |
7 | 6 | shftfval 13658 | . . 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 5372 | . 2 ⊢ (𝐹 ↾ ℂ) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑥𝐹𝑦)} | |
10 | 4, 8, 9 | 3eqtr4i 2642 | 1 ⊢ (𝐹 shift 0) = (𝐹 ↾ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 {copab 4642 ↾ cres 5040 (class class class)co 6549 ℂcc 9813 0cc0 9815 − cmin 10145 shift cshi 13654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-shft 13655 |
This theorem is referenced by: shftidt 13670 |
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