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Mirrors > Home > ILE Home > Th. List > nummul1c | Unicode version |
Description: The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
nummul1c.1 | |
nummul1c.2 | |
nummul1c.3 | |
nummul1c.4 | |
nummul1c.5 | |
nummul1c.6 | |
nummul1c.7 | |
nummul1c.8 | |
nummul1c.9 |
Ref | Expression |
---|---|
nummul1c |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nummul1c.5 | . . . 4 | |
2 | nummul1c.1 | . . . . 5 | |
3 | nummul1c.3 | . . . . 5 | |
4 | nummul1c.4 | . . . . 5 | |
5 | 2, 3, 4 | numcl 9348 | . . . 4 |
6 | 1, 5 | eqeltri 2243 | . . 3 |
7 | nummul1c.2 | . . 3 | |
8 | 6, 7 | num0u 9346 | . 2 |
9 | 0nn0 9143 | . . 3 | |
10 | 2, 9 | num0h 9347 | . . 3 |
11 | nummul1c.6 | . . 3 | |
12 | nummul1c.7 | . . 3 | |
13 | 12 | nn0cni 9140 | . . . . . 6 |
14 | 13 | addid2i 8055 | . . . . 5 |
15 | 14 | oveq2i 5862 | . . . 4 |
16 | nummul1c.8 | . . . 4 | |
17 | 15, 16 | eqtri 2191 | . . 3 |
18 | 4, 7 | num0u 9346 | . . . 4 |
19 | nummul1c.9 | . . . 4 | |
20 | 18, 19 | eqtr3i 2193 | . . 3 |
21 | 2, 3, 4, 9, 9, 1, 10, 7, 11, 12, 17, 20 | nummac 9380 | . 2 |
22 | 8, 21 | eqtri 2191 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1348 wcel 2141 (class class class)co 5851 cc0 7767 caddc 7770 cmul 7772 cn0 9128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-cnre 7878 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-sub 8085 df-inn 8872 df-n0 9129 |
This theorem is referenced by: nummul2c 9385 decmul1 9399 decmul1c 9400 |
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