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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 3753 | . 2 ⊢ (𝑋 ∈ (ℕ0 ∪ {-∞}) ↔ (𝑋 ∈ ℕ0 ∨ 𝑋 ∈ {-∞})) | |
2 | nn0z 11400 | . . . 4 ⊢ (𝑋 ∈ ℕ0 → 𝑋 ∈ ℤ) | |
3 | zltlem1 11430 | . . . 4 ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑋 < 𝑌 ↔ 𝑋 ≤ (𝑌 − 1))) | |
4 | 2, 3 | sylan 488 | . . 3 ⊢ ((𝑋 ∈ ℕ0 ∧ 𝑌 ∈ ℤ) → (𝑋 < 𝑌 ↔ 𝑋 ≤ (𝑌 − 1))) |
5 | zre 11381 | . . . . . . 7 ⊢ (𝑌 ∈ ℤ → 𝑌 ∈ ℝ) | |
6 | mnflt 11957 | . . . . . . 7 ⊢ (𝑌 ∈ ℝ → -∞ < 𝑌) | |
7 | 5, 6 | syl 17 | . . . . . 6 ⊢ (𝑌 ∈ ℤ → -∞ < 𝑌) |
8 | peano2zm 11420 | . . . . . . . . 9 ⊢ (𝑌 ∈ ℤ → (𝑌 − 1) ∈ ℤ) | |
9 | 8 | zred 11482 | . . . . . . . 8 ⊢ (𝑌 ∈ ℤ → (𝑌 − 1) ∈ ℝ) |
10 | 9 | rexrd 10089 | . . . . . . 7 ⊢ (𝑌 ∈ ℤ → (𝑌 − 1) ∈ ℝ*) |
11 | mnfle 11969 | . . . . . . 7 ⊢ ((𝑌 − 1) ∈ ℝ* → -∞ ≤ (𝑌 − 1)) | |
12 | 10, 11 | syl 17 | . . . . . 6 ⊢ (𝑌 ∈ ℤ → -∞ ≤ (𝑌 − 1)) |
13 | 7, 12 | 2thd 255 | . . . . 5 ⊢ (𝑌 ∈ ℤ → (-∞ < 𝑌 ↔ -∞ ≤ (𝑌 − 1))) |
14 | elsni 4194 | . . . . . 6 ⊢ (𝑋 ∈ {-∞} → 𝑋 = -∞) | |
15 | breq1 4656 | . . . . . . 7 ⊢ (𝑋 = -∞ → (𝑋 < 𝑌 ↔ -∞ < 𝑌)) | |
16 | breq1 4656 | . . . . . . 7 ⊢ (𝑋 = -∞ → (𝑋 ≤ (𝑌 − 1) ↔ -∞ ≤ (𝑌 − 1))) | |
17 | 15, 16 | bibi12d 335 | . . . . . 6 ⊢ (𝑋 = -∞ → ((𝑋 < 𝑌 ↔ 𝑋 ≤ (𝑌 − 1)) ↔ (-∞ < 𝑌 ↔ -∞ ≤ (𝑌 − 1)))) |
18 | 14, 17 | syl 17 | . . . . 5 ⊢ (𝑋 ∈ {-∞} → ((𝑋 < 𝑌 ↔ 𝑋 ≤ (𝑌 − 1)) ↔ (-∞ < 𝑌 ↔ -∞ ≤ (𝑌 − 1)))) |
19 | 13, 18 | syl5ibrcom 237 | . . . 4 ⊢ (𝑌 ∈ ℤ → (𝑋 ∈ {-∞} → (𝑋 < 𝑌 ↔ 𝑋 ≤ (𝑌 − 1)))) |
20 | 19 | impcom 446 | . . 3 ⊢ ((𝑋 ∈ {-∞} ∧ 𝑌 ∈ ℤ) → (𝑋 < 𝑌 ↔ 𝑋 ≤ (𝑌 − 1))) |
21 | 4, 20 | jaoian 824 | . 2 ⊢ (((𝑋 ∈ ℕ0 ∨ 𝑋 ∈ {-∞}) ∧ 𝑌 ∈ ℤ) → (𝑋 < 𝑌 ↔ 𝑋 ≤ (𝑌 − 1))) |
22 | 1, 21 | sylanb 489 | 1 ⊢ ((𝑋 ∈ (ℕ0 ∪ {-∞}) ∧ 𝑌 ∈ ℤ) → (𝑋 < 𝑌 ↔ 𝑋 ≤ (𝑌 − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∪ cun 3572 {csn 4177 class class class wbr 4653 (class class class)co 6650 ℝcr 9935 1c1 9937 -∞cmnf 10072 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 − cmin 10266 ℕ0cn0 11292 ℤcz 11377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 |
This theorem is referenced by: degltp1le 23833 ply1divex 23896 |
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