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Mirrors > Home > ILE Home > Th. List > 3declth | GIF version |
Description: Comparing two decimal integers with three "digits" (unequal higher places). (Contributed by AV, 8-Sep-2021.) |
Ref | Expression |
---|---|
3decltc.a | ⊢ 𝐴 ∈ ℕ0 |
3decltc.b | ⊢ 𝐵 ∈ ℕ0 |
3decltc.c | ⊢ 𝐶 ∈ ℕ0 |
3decltc.d | ⊢ 𝐷 ∈ ℕ0 |
3decltc.e | ⊢ 𝐸 ∈ ℕ0 |
3decltc.f | ⊢ 𝐹 ∈ ℕ0 |
3decltc.3 | ⊢ 𝐴 < 𝐵 |
3declth.1 | ⊢ 𝐶 ≤ 9 |
3declth.2 | ⊢ 𝐸 ≤ 9 |
Ref | Expression |
---|---|
3declth | ⊢ ;;𝐴𝐶𝐸 < ;;𝐵𝐷𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3decltc.a | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
2 | 3decltc.c | . . 3 ⊢ 𝐶 ∈ ℕ0 | |
3 | 1, 2 | deccl 9428 | . 2 ⊢ ;𝐴𝐶 ∈ ℕ0 |
4 | 3decltc.b | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
5 | 3decltc.d | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
6 | 4, 5 | deccl 9428 | . 2 ⊢ ;𝐵𝐷 ∈ ℕ0 |
7 | 3decltc.e | . 2 ⊢ 𝐸 ∈ ℕ0 | |
8 | 3decltc.f | . 2 ⊢ 𝐹 ∈ ℕ0 | |
9 | 3declth.2 | . 2 ⊢ 𝐸 ≤ 9 | |
10 | 3declth.1 | . . 3 ⊢ 𝐶 ≤ 9 | |
11 | 3decltc.3 | . . 3 ⊢ 𝐴 < 𝐵 | |
12 | 1, 4, 2, 5, 10, 11 | declth 9443 | . 2 ⊢ ;𝐴𝐶 < ;𝐵𝐷 |
13 | 3, 6, 7, 8, 9, 12 | declth 9443 | 1 ⊢ ;;𝐴𝐶𝐸 < ;;𝐵𝐷𝐹 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 class class class wbr 4018 < clt 8022 ≤ cle 8023 9c9 9007 ℕ0cn0 9206 ;cdc 9414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-mulrcl 7940 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-1rid 7948 ax-0id 7949 ax-rnegex 7950 ax-precex 7951 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-ltadd 7957 ax-pre-mulgt0 7958 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-inn 8950 df-2 9008 df-3 9009 df-4 9010 df-5 9011 df-6 9012 df-7 9013 df-8 9014 df-9 9015 df-n0 9207 df-z 9284 df-dec 9415 |
This theorem is referenced by: (None) |
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