![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 4t3lem | Structured version Visualization version GIF version |
Description: Lemma for 4t3e12 12856 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 7459 | . 2 ⊢ (𝐴 · 𝐶) = (𝐴 · (𝐵 + 1)) |
3 | 4t3lem.1 | . . . . . 6 ⊢ 𝐴 ∈ ℕ0 | |
4 | 3 | nn0cni 12565 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
5 | 4t3lem.2 | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
6 | 5 | nn0cni 12565 | . . . . 5 ⊢ 𝐵 ∈ ℂ |
7 | ax-1cn 11242 | . . . . 5 ⊢ 1 ∈ ℂ | |
8 | 4, 6, 7 | adddii 11302 | . . . 4 ⊢ (𝐴 · (𝐵 + 1)) = ((𝐴 · 𝐵) + (𝐴 · 1)) |
9 | 4t3lem.4 | . . . . 5 ⊢ (𝐴 · 𝐵) = 𝐷 | |
10 | 4 | mulridi 11294 | . . . . 5 ⊢ (𝐴 · 1) = 𝐴 |
11 | 9, 10 | oveq12i 7460 | . . . 4 ⊢ ((𝐴 · 𝐵) + (𝐴 · 1)) = (𝐷 + 𝐴) |
12 | 8, 11 | eqtri 2768 | . . 3 ⊢ (𝐴 · (𝐵 + 1)) = (𝐷 + 𝐴) |
13 | 4t3lem.5 | . . 3 ⊢ (𝐷 + 𝐴) = 𝐸 | |
14 | 12, 13 | eqtri 2768 | . 2 ⊢ (𝐴 · (𝐵 + 1)) = 𝐸 |
15 | 2, 14 | eqtri 2768 | 1 ⊢ (𝐴 · 𝐶) = 𝐸 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2108 (class class class)co 7448 1c1 11185 + caddc 11187 · cmul 11189 ℕ0cn0 12553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-mulcl 11246 ax-mulcom 11248 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1rid 11254 ax-cnre 11257 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-om 7904 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-nn 12294 df-n0 12554 |
This theorem is referenced by: 4t3e12 12856 4t4e16 12857 5t2e10 12858 5t3e15 12859 5t4e20 12860 5t5e25 12861 6t3e18 12863 6t4e24 12864 6t5e30 12865 6t6e36 12866 7t3e21 12868 7t4e28 12869 7t5e35 12870 7t6e42 12871 7t7e49 12872 8t3e24 12874 8t4e32 12875 8t5e40 12876 8t6e48 12877 8t7e56 12878 8t8e64 12879 9t3e27 12881 9t4e36 12882 9t5e45 12883 9t6e54 12884 9t7e63 12885 9t8e72 12886 9t9e81 12887 |
Copyright terms: Public domain | W3C validator |