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Mirrors > Home > ILE Home > Th. List > 4t3lem | GIF version |
Description: Lemma for 4t3e12 9467 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
4t3lem.1 | ⊢ 𝐴 ∈ ℕ0 |
4t3lem.2 | ⊢ 𝐵 ∈ ℕ0 |
4t3lem.3 | ⊢ 𝐶 = (𝐵 + 1) |
4t3lem.4 | ⊢ (𝐴 · 𝐵) = 𝐷 |
4t3lem.5 | ⊢ (𝐷 + 𝐴) = 𝐸 |
Ref | Expression |
---|---|
4t3lem | ⊢ (𝐴 · 𝐶) = 𝐸 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4t3lem.3 | . . 3 ⊢ 𝐶 = (𝐵 + 1) | |
2 | 1 | oveq2i 5880 | . 2 ⊢ (𝐴 · 𝐶) = (𝐴 · (𝐵 + 1)) |
3 | 4t3lem.1 | . . . . . 6 ⊢ 𝐴 ∈ ℕ0 | |
4 | 3 | nn0cni 9174 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
5 | 4t3lem.2 | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
6 | 5 | nn0cni 9174 | . . . . 5 ⊢ 𝐵 ∈ ℂ |
7 | ax-1cn 7892 | . . . . 5 ⊢ 1 ∈ ℂ | |
8 | 4, 6, 7 | adddii 7955 | . . . 4 ⊢ (𝐴 · (𝐵 + 1)) = ((𝐴 · 𝐵) + (𝐴 · 1)) |
9 | 4t3lem.4 | . . . . 5 ⊢ (𝐴 · 𝐵) = 𝐷 | |
10 | 4 | mulid1i 7947 | . . . . 5 ⊢ (𝐴 · 1) = 𝐴 |
11 | 9, 10 | oveq12i 5881 | . . . 4 ⊢ ((𝐴 · 𝐵) + (𝐴 · 1)) = (𝐷 + 𝐴) |
12 | 8, 11 | eqtri 2198 | . . 3 ⊢ (𝐴 · (𝐵 + 1)) = (𝐷 + 𝐴) |
13 | 4t3lem.5 | . . 3 ⊢ (𝐷 + 𝐴) = 𝐸 | |
14 | 12, 13 | eqtri 2198 | . 2 ⊢ (𝐴 · (𝐵 + 1)) = 𝐸 |
15 | 2, 14 | eqtri 2198 | 1 ⊢ (𝐴 · 𝐶) = 𝐸 |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 (class class class)co 5869 1c1 7800 + caddc 7802 · cmul 7804 ℕ0cn0 9162 |
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-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-sep 4118 ax-cnex 7890 ax-resscn 7891 ax-1cn 7892 ax-1re 7893 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-mulcom 7900 ax-mulass 7902 ax-distr 7903 ax-1rid 7906 ax-rnegex 7908 ax-cnre 7910 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-iota 5174 df-fv 5220 df-ov 5872 df-inn 8906 df-n0 9163 |
This theorem is referenced by: 4t3e12 9467 4t4e16 9468 5t2e10 9469 5t3e15 9470 5t4e20 9471 5t5e25 9472 6t3e18 9474 6t4e24 9475 6t5e30 9476 6t6e36 9477 7t3e21 9479 7t4e28 9480 7t5e35 9481 7t6e42 9482 7t7e49 9483 8t3e24 9485 8t4e32 9486 8t5e40 9487 8t6e48 9488 8t7e56 9489 8t8e64 9490 9t3e27 9492 9t4e36 9493 9t5e45 9494 9t6e54 9495 9t7e63 9496 9t8e72 9497 9t9e81 9498 |
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