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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 9048 | . 2 ⊢ ;𝐴𝐶 ∈ ℕ0 |
4 | 3decltc.b | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
5 | 3decltc.d | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
6 | 4, 5 | deccl 9048 | . 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 9063 | . 2 ⊢ ;𝐴𝐶 < ;𝐵𝐷 |
13 | 3, 6, 7, 8, 9, 12 | declth 9063 | 1 ⊢ ;;𝐴𝐶𝐸 < ;;𝐵𝐷𝐹 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1448 class class class wbr 3875 < clt 7672 ≤ cle 7673 9c9 8636 ℕ0cn0 8829 ;cdc 9034 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-precex 7605 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-ltadd 7611 ax-pre-mulgt0 7612 |
This theorem depends on definitions: df-bi 116 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-br 3876 df-opab 3930 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-iota 5024 df-fun 5061 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-inn 8579 df-2 8637 df-3 8638 df-4 8639 df-5 8640 df-6 8641 df-7 8642 df-8 8643 df-9 8644 df-n0 8830 df-z 8907 df-dec 9035 |
This theorem is referenced by: (None) |
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