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Mirrors > Home > MPE Home > Th. List > decleh | Structured version Visualization version GIF version |
Description: Comparing two decimal integers (unequal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.) |
Ref | Expression |
---|---|
decle.1 | ⊢ 𝐴 ∈ ℕ0 |
decle.2 | ⊢ 𝐵 ∈ ℕ0 |
decle.3 | ⊢ 𝐶 ∈ ℕ0 |
decleh.4 | ⊢ 𝐷 ∈ ℕ0 |
decleh.5 | ⊢ 𝐶 ≤ 9 |
decleh.6 | ⊢ 𝐴 < 𝐵 |
Ref | Expression |
---|---|
decleh | ⊢ ;𝐴𝐶 ≤ ;𝐵𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decle.1 | . . . 4 ⊢ 𝐴 ∈ ℕ0 | |
2 | decle.3 | . . . 4 ⊢ 𝐶 ∈ ℕ0 | |
3 | 1, 2 | deccl 12723 | . . 3 ⊢ ;𝐴𝐶 ∈ ℕ0 |
4 | 3 | nn0rei 12514 | . 2 ⊢ ;𝐴𝐶 ∈ ℝ |
5 | decle.2 | . . . 4 ⊢ 𝐵 ∈ ℕ0 | |
6 | decleh.4 | . . . 4 ⊢ 𝐷 ∈ ℕ0 | |
7 | 5, 6 | deccl 12723 | . . 3 ⊢ ;𝐵𝐷 ∈ ℕ0 |
8 | 7 | nn0rei 12514 | . 2 ⊢ ;𝐵𝐷 ∈ ℝ |
9 | decleh.5 | . . 3 ⊢ 𝐶 ≤ 9 | |
10 | decleh.6 | . . 3 ⊢ 𝐴 < 𝐵 | |
11 | 1, 5, 2, 6, 9, 10 | declth 12738 | . 2 ⊢ ;𝐴𝐶 < ;𝐵𝐷 |
12 | 4, 8, 11 | ltleii 11368 | 1 ⊢ ;𝐴𝐶 ≤ ;𝐵𝐷 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 class class class wbr 5148 < clt 11279 ≤ cle 11280 9c9 12305 ℕ0cn0 12503 ;cdc 12708 |
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-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-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 |
This theorem is referenced by: declei 12744 3lexlogpow5ineq2 41526 3lexlogpow5ineq5 41531 31prm 46937 nfermltl2rev 47083 |
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