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Mirrors > Home > MPE Home > Th. List > degltlem1 | Structured version Visualization version GIF version |
Description: Theorem on arithmetic of extended reals useful for degrees. (Contributed by Stefan O'Rear, 23-Mar-2015.) |
Ref | Expression |
---|---|
degltlem1 | ⊢ ((𝑋 ∈ (ℕ0 ∪ {-∞}) ∧ 𝑌 ∈ ℤ) → (𝑋 < 𝑌 ↔ 𝑋 ≤ (𝑌 − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elun 4143 | . 2 ⊢ (𝑋 ∈ (ℕ0 ∪ {-∞}) ↔ (𝑋 ∈ ℕ0 ∨ 𝑋 ∈ {-∞})) | |
2 | nn0z 12587 | . . . 4 ⊢ (𝑋 ∈ ℕ0 → 𝑋 ∈ ℤ) | |
3 | zltlem1 12619 | . . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑋 < 𝑌 ↔ 𝑋 ≤ (𝑌 − 1))) | |
4 | 2, 3 | sylan 579 | . . 3 ⊢ ((𝑋 ∈ ℕ0 ∧ 𝑌 ∈ ℤ) → (𝑋 < 𝑌 ↔ 𝑋 ≤ (𝑌 − 1))) |
5 | zre 12566 | . . . . . . 7 ⊢ (𝑌 ∈ ℤ → 𝑌 ∈ ℝ) | |
6 | 5 | mnfltd 13110 | . . . . . 6 ⊢ (𝑌 ∈ ℤ → -∞ < 𝑌) |
7 | peano2zm 12609 | . . . . . . . . 9 ⊢ (𝑌 ∈ ℤ → (𝑌 − 1) ∈ ℤ) | |
8 | 7 | zred 12670 | . . . . . . . 8 ⊢ (𝑌 ∈ ℤ → (𝑌 − 1) ∈ ℝ) |
9 | 8 | rexrd 11268 | . . . . . . 7 ⊢ (𝑌 ∈ ℤ → (𝑌 − 1) ∈ ℝ*) |
10 | mnfle 13120 | . . . . . . 7 ⊢ ((𝑌 − 1) ∈ ℝ* → -∞ ≤ (𝑌 − 1)) | |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ (𝑌 ∈ ℤ → -∞ ≤ (𝑌 − 1)) |
12 | 6, 11 | 2thd 265 | . . . . 5 ⊢ (𝑌 ∈ ℤ → (-∞ < 𝑌 ↔ -∞ ≤ (𝑌 − 1))) |
13 | elsni 4640 | . . . . . 6 ⊢ (𝑋 ∈ {-∞} → 𝑋 = -∞) | |
14 | breq1 5144 | . . . . . . 7 ⊢ (𝑋 = -∞ → (𝑋 < 𝑌 ↔ -∞ < 𝑌)) | |
15 | breq1 5144 | . . . . . . 7 ⊢ (𝑋 = -∞ → (𝑋 ≤ (𝑌 − 1) ↔ -∞ ≤ (𝑌 − 1))) | |
16 | 14, 15 | bibi12d 345 | . . . . . 6 ⊢ (𝑋 = -∞ → ((𝑋 < 𝑌 ↔ 𝑋 ≤ (𝑌 − 1)) ↔ (-∞ < 𝑌 ↔ -∞ ≤ (𝑌 − 1)))) |
17 | 13, 16 | syl 17 | . . . . 5 ⊢ (𝑋 ∈ {-∞} → ((𝑋 < 𝑌 ↔ 𝑋 ≤ (𝑌 − 1)) ↔ (-∞ < 𝑌 ↔ -∞ ≤ (𝑌 − 1)))) |
18 | 12, 17 | syl5ibrcom 246 | . . . 4 ⊢ (𝑌 ∈ ℤ → (𝑋 ∈ {-∞} → (𝑋 < 𝑌 ↔ 𝑋 ≤ (𝑌 − 1)))) |
19 | 18 | impcom 407 | . . 3 ⊢ ((𝑋 ∈ {-∞} ∧ 𝑌 ∈ ℤ) → (𝑋 < 𝑌 ↔ 𝑋 ≤ (𝑌 − 1))) |
20 | 4, 19 | jaoian 953 | . 2 ⊢ (((𝑋 ∈ ℕ0 ∨ 𝑋 ∈ {-∞}) ∧ 𝑌 ∈ ℤ) → (𝑋 < 𝑌 ↔ 𝑋 ≤ (𝑌 − 1))) |
21 | 1, 20 | sylanb 580 | 1 ⊢ ((𝑋 ∈ (ℕ0 ∪ {-∞}) ∧ 𝑌 ∈ ℤ) → (𝑋 < 𝑌 ↔ 𝑋 ≤ (𝑌 − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ∪ cun 3941 {csn 4623 class class class wbr 5141 (class class class)co 7405 1c1 11113 -∞cmnf 11250 ℝ*cxr 11251 < clt 11252 ≤ cle 11253 − cmin 11448 ℕ0cn0 12476 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-cnex 11168 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 ax-pre-mulgt0 11189 |
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-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-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 |
This theorem is referenced by: degltp1le 25964 ply1divex 26027 algextdeglem8 33301 |
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