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Mirrors > Home > MPE Home > Th. List > shftuz | Structured version Visualization version GIF version |
Description: A shift of the upper integers. (Contributed by Mario Carneiro, 5-Nov-2013.) |
Ref | Expression |
---|---|
shftuz | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ (ℤ≥‘𝐵)} = (ℤ≥‘(𝐵 + 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3430 | . 2 ⊢ {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ (ℤ≥‘𝐵)} = {𝑥 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴) ∈ (ℤ≥‘𝐵))} | |
2 | simp2 1135 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑥 ∈ ℂ ∧ (𝑥 − 𝐴) ∈ (ℤ≥‘𝐵)) → 𝑥 ∈ ℂ) | |
3 | zcn 12594 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
4 | 3 | 3ad2ant1 1131 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑥 ∈ ℂ ∧ (𝑥 − 𝐴) ∈ (ℤ≥‘𝐵)) → 𝐴 ∈ ℂ) |
5 | 2, 4 | npcand 11606 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑥 ∈ ℂ ∧ (𝑥 − 𝐴) ∈ (ℤ≥‘𝐵)) → ((𝑥 − 𝐴) + 𝐴) = 𝑥) |
6 | eluzadd 12882 | . . . . . . . . 9 ⊢ (((𝑥 − 𝐴) ∈ (ℤ≥‘𝐵) ∧ 𝐴 ∈ ℤ) → ((𝑥 − 𝐴) + 𝐴) ∈ (ℤ≥‘(𝐵 + 𝐴))) | |
7 | 6 | ancoms 458 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝑥 − 𝐴) ∈ (ℤ≥‘𝐵)) → ((𝑥 − 𝐴) + 𝐴) ∈ (ℤ≥‘(𝐵 + 𝐴))) |
8 | 7 | 3adant2 1129 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑥 ∈ ℂ ∧ (𝑥 − 𝐴) ∈ (ℤ≥‘𝐵)) → ((𝑥 − 𝐴) + 𝐴) ∈ (ℤ≥‘(𝐵 + 𝐴))) |
9 | 5, 8 | eqeltrrd 2830 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑥 ∈ ℂ ∧ (𝑥 − 𝐴) ∈ (ℤ≥‘𝐵)) → 𝑥 ∈ (ℤ≥‘(𝐵 + 𝐴))) |
10 | 9 | 3expib 1120 | . . . . 5 ⊢ (𝐴 ∈ ℤ → ((𝑥 ∈ ℂ ∧ (𝑥 − 𝐴) ∈ (ℤ≥‘𝐵)) → 𝑥 ∈ (ℤ≥‘(𝐵 + 𝐴)))) |
11 | 10 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑥 ∈ ℂ ∧ (𝑥 − 𝐴) ∈ (ℤ≥‘𝐵)) → 𝑥 ∈ (ℤ≥‘(𝐵 + 𝐴)))) |
12 | eluzelcn 12865 | . . . . . 6 ⊢ (𝑥 ∈ (ℤ≥‘(𝐵 + 𝐴)) → 𝑥 ∈ ℂ) | |
13 | 12 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑥 ∈ (ℤ≥‘(𝐵 + 𝐴)) → 𝑥 ∈ ℂ)) |
14 | eluzsub 12883 | . . . . . . 7 ⊢ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝑥 ∈ (ℤ≥‘(𝐵 + 𝐴))) → (𝑥 − 𝐴) ∈ (ℤ≥‘𝐵)) | |
15 | 14 | 3expia 1119 | . . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝑥 ∈ (ℤ≥‘(𝐵 + 𝐴)) → (𝑥 − 𝐴) ∈ (ℤ≥‘𝐵))) |
16 | 15 | ancoms 458 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑥 ∈ (ℤ≥‘(𝐵 + 𝐴)) → (𝑥 − 𝐴) ∈ (ℤ≥‘𝐵))) |
17 | 13, 16 | jcad 512 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑥 ∈ (ℤ≥‘(𝐵 + 𝐴)) → (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴) ∈ (ℤ≥‘𝐵)))) |
18 | 11, 17 | impbid 211 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑥 ∈ ℂ ∧ (𝑥 − 𝐴) ∈ (ℤ≥‘𝐵)) ↔ 𝑥 ∈ (ℤ≥‘(𝐵 + 𝐴)))) |
19 | 18 | eqabcdv 2864 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝑥 ∣ (𝑥 ∈ ℂ ∧ (𝑥 − 𝐴) ∈ (ℤ≥‘𝐵))} = (ℤ≥‘(𝐵 + 𝐴))) |
20 | 1, 19 | eqtrid 2780 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝑥 ∈ ℂ ∣ (𝑥 − 𝐴) ∈ (ℤ≥‘𝐵)} = (ℤ≥‘(𝐵 + 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 {cab 2705 {crab 3429 ‘cfv 6548 (class class class)co 7420 ℂcc 11137 + caddc 11142 − cmin 11475 ℤcz 12589 ℤ≥cuz 12853 |
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-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
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-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 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-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 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-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 |
This theorem is referenced by: seqshft 15065 uzmptshftfval 43783 |
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