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Mirrors > Home > MPE Home > Th. List > zriotaneg | Structured version Visualization version GIF version |
Description: The negative of the unique integer such that 𝜑. (Contributed by AV, 1-Dec-2018.) |
Ref | Expression |
---|---|
zriotaneg.1 | ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
zriotaneg | ⊢ (∃!𝑥 ∈ ℤ 𝜑 → (℩𝑥 ∈ ℤ 𝜑) = -(℩𝑦 ∈ ℤ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1537 | . 2 ⊢ ⊤ | |
2 | nfriota1 7368 | . . . 4 ⊢ Ⅎ𝑦(℩𝑦 ∈ ℤ 𝜓) | |
3 | 2 | nfneg 11460 | . . 3 ⊢ Ⅎ𝑦-(℩𝑦 ∈ ℤ 𝜓) |
4 | znegcl 12601 | . . . 4 ⊢ (𝑦 ∈ ℤ → -𝑦 ∈ ℤ) | |
5 | 4 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℤ) → -𝑦 ∈ ℤ) |
6 | simpr 484 | . . . 4 ⊢ ((⊤ ∧ (℩𝑦 ∈ ℤ 𝜓) ∈ ℤ) → (℩𝑦 ∈ ℤ 𝜓) ∈ ℤ) | |
7 | 6 | znegcld 12672 | . . 3 ⊢ ((⊤ ∧ (℩𝑦 ∈ ℤ 𝜓) ∈ ℤ) → -(℩𝑦 ∈ ℤ 𝜓) ∈ ℤ) |
8 | zriotaneg.1 | . . 3 ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜓)) | |
9 | negeq 11456 | . . 3 ⊢ (𝑦 = (℩𝑦 ∈ ℤ 𝜓) → -𝑦 = -(℩𝑦 ∈ ℤ 𝜓)) | |
10 | znegcl 12601 | . . . . 5 ⊢ (𝑥 ∈ ℤ → -𝑥 ∈ ℤ) | |
11 | zcn 12567 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
12 | zcn 12567 | . . . . . 6 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ) | |
13 | negcon2 11517 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥)) | |
14 | 11, 12, 13 | syl2an 595 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥)) |
15 | 10, 14 | reuhyp 5411 | . . . 4 ⊢ (𝑥 ∈ ℤ → ∃!𝑦 ∈ ℤ 𝑥 = -𝑦) |
16 | 15 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℤ) → ∃!𝑦 ∈ ℤ 𝑥 = -𝑦) |
17 | 3, 5, 7, 8, 9, 16 | riotaxfrd 7396 | . 2 ⊢ ((⊤ ∧ ∃!𝑥 ∈ ℤ 𝜑) → (℩𝑥 ∈ ℤ 𝜑) = -(℩𝑦 ∈ ℤ 𝜓)) |
18 | 1, 17 | mpan 687 | 1 ⊢ (∃!𝑥 ∈ ℤ 𝜑 → (℩𝑥 ∈ ℤ 𝜑) = -(℩𝑦 ∈ ℤ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ∃!wreu 3368 ℩crio 7360 ℂcc 11110 -cneg 11449 ℤcz 12562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-ltxr 11257 df-sub 11450 df-neg 11451 df-nn 12217 df-z 12563 |
This theorem is referenced by: dfceil2 13810 |
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