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Mirrors > Home > MPE Home > Th. List > declecOLD | Structured version Visualization version GIF version |
Description: Obsolete version of decleh 11579 as of 8-Sep-2021. (Contributed by AV, 17-Aug-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
decleOLD.1 | ⊢ 𝐴 ∈ ℕ0 |
decleOLD.2 | ⊢ 𝐵 ∈ ℕ0 |
decleOLD.3 | ⊢ 𝐶 ∈ ℕ0 |
declecOLD.4 | ⊢ 𝐷 ∈ ℕ0 |
declecOLD.5 | ⊢ 𝐶 ≤ 9 |
declecOLD.6 | ⊢ 𝐴 < 𝐵 |
Ref | Expression |
---|---|
declecOLD | ⊢ ;𝐴𝐶 ≤ ;𝐵𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decleOLD.1 | . . . 4 ⊢ 𝐴 ∈ ℕ0 | |
2 | decleOLD.3 | . . . 4 ⊢ 𝐶 ∈ ℕ0 | |
3 | 1, 2 | deccl 11550 | . . 3 ⊢ ;𝐴𝐶 ∈ ℕ0 |
4 | 3 | nn0rei 11341 | . 2 ⊢ ;𝐴𝐶 ∈ ℝ |
5 | decleOLD.2 | . . . 4 ⊢ 𝐵 ∈ ℕ0 | |
6 | declecOLD.4 | . . . 4 ⊢ 𝐷 ∈ ℕ0 | |
7 | 5, 6 | deccl 11550 | . . 3 ⊢ ;𝐵𝐷 ∈ ℕ0 |
8 | 7 | nn0rei 11341 | . 2 ⊢ ;𝐵𝐷 ∈ ℝ |
9 | declecOLD.5 | . . . 4 ⊢ 𝐶 ≤ 9 | |
10 | 2 | nn0zi 11440 | . . . . . 6 ⊢ 𝐶 ∈ ℤ |
11 | 9nn0 11354 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
12 | 11 | nn0zi 11440 | . . . . . 6 ⊢ 9 ∈ ℤ |
13 | zleltp1 11466 | . . . . . 6 ⊢ ((𝐶 ∈ ℤ ∧ 9 ∈ ℤ) → (𝐶 ≤ 9 ↔ 𝐶 < (9 + 1))) | |
14 | 10, 12, 13 | mp2an 708 | . . . . 5 ⊢ (𝐶 ≤ 9 ↔ 𝐶 < (9 + 1)) |
15 | 9p1e10OLD 11197 | . . . . . 6 ⊢ (9 + 1) = 10 | |
16 | 15 | breq2i 4693 | . . . . 5 ⊢ (𝐶 < (9 + 1) ↔ 𝐶 < 10) |
17 | 14, 16 | bitri 264 | . . . 4 ⊢ (𝐶 ≤ 9 ↔ 𝐶 < 10) |
18 | 9, 17 | mpbi 220 | . . 3 ⊢ 𝐶 < 10 |
19 | declecOLD.6 | . . 3 ⊢ 𝐴 < 𝐵 | |
20 | 1, 5, 2, 6, 18, 19 | decltcOLD 11571 | . 2 ⊢ ;𝐴𝐶 < ;𝐵𝐷 |
21 | 4, 8, 20 | ltleii 10198 | 1 ⊢ ;𝐴𝐶 ≤ ;𝐵𝐷 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∈ wcel 2030 class class class wbr 4685 (class class class)co 6690 1c1 9975 + caddc 9977 < clt 10112 ≤ cle 10113 9c9 11115 10c10 11116 ℕ0cn0 11330 ℤcz 11415 ;cdc 11531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-10OLD 11125 df-n0 11331 df-z 11416 df-dec 11532 |
This theorem is referenced by: (None) |
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