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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 11510 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥 − 0) = 𝑥) | |
2 | 1 | breq1d 5158 | . . . 4 ⊢ (𝑥 ∈ ℂ → ((𝑥 − 0)𝐹𝑦 ↔ 𝑥𝐹𝑦)) |
3 | 2 | pm5.32i 574 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ (𝑥 − 0)𝐹𝑦) ↔ (𝑥 ∈ ℂ ∧ 𝑥𝐹𝑦)) |
4 | 3 | opabbii 5215 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 0)𝐹𝑦)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑥𝐹𝑦)} |
5 | 0cn 11236 | . . 3 ⊢ 0 ∈ ℂ | |
6 | shftfval.1 | . . . 4 ⊢ 𝐹 ∈ V | |
7 | 6 | shftfval 15049 | . . 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 6045 | . 2 ⊢ (𝐹 ↾ ℂ) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℂ ∧ 𝑥𝐹𝑦)} | |
10 | 4, 8, 9 | 3eqtr4i 2766 | 1 ⊢ (𝐹 shift 0) = (𝐹 ↾ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 class class class wbr 5148 {copab 5210 ↾ cres 5680 (class class class)co 7420 ℂcc 11136 0cc0 11138 − cmin 11474 shift cshi 15045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 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 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-ltxr 11283 df-sub 11476 df-shft 15046 |
This theorem is referenced by: shftidt 15061 |
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