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Mirrors > Home > MPE Home > Th. List > 4t3lem | Structured version Visualization version GIF version |
Description: Lemma for 4t3e12 11508 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 6560 | . 2 ⊢ (𝐴 · 𝐶) = (𝐴 · (𝐵 + 1)) |
3 | 4t3lem.1 | . . . . . 6 ⊢ 𝐴 ∈ ℕ0 | |
4 | 3 | nn0cni 11181 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
5 | 4t3lem.2 | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
6 | 5 | nn0cni 11181 | . . . . 5 ⊢ 𝐵 ∈ ℂ |
7 | ax-1cn 9873 | . . . . 5 ⊢ 1 ∈ ℂ | |
8 | 4, 6, 7 | adddii 9929 | . . . 4 ⊢ (𝐴 · (𝐵 + 1)) = ((𝐴 · 𝐵) + (𝐴 · 1)) |
9 | 4t3lem.4 | . . . . 5 ⊢ (𝐴 · 𝐵) = 𝐷 | |
10 | 4 | mulid1i 9921 | . . . . 5 ⊢ (𝐴 · 1) = 𝐴 |
11 | 9, 10 | oveq12i 6561 | . . . 4 ⊢ ((𝐴 · 𝐵) + (𝐴 · 1)) = (𝐷 + 𝐴) |
12 | 8, 11 | eqtri 2632 | . . 3 ⊢ (𝐴 · (𝐵 + 1)) = (𝐷 + 𝐴) |
13 | 4t3lem.5 | . . 3 ⊢ (𝐷 + 𝐴) = 𝐸 | |
14 | 12, 13 | eqtri 2632 | . 2 ⊢ (𝐴 · (𝐵 + 1)) = 𝐸 |
15 | 2, 14 | eqtri 2632 | 1 ⊢ (𝐴 · 𝐶) = 𝐸 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 (class class class)co 6549 1c1 9816 + caddc 9818 · cmul 9820 ℕ0cn0 11169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 df-n0 11170 |
This theorem is referenced by: 4t3e12 11508 4t4e16 11509 5t2e10 11510 5t3e15 11511 5t3e15OLD 11512 5t4e20 11513 5t4e20OLD 11514 5t5e25 11515 5t5e25OLD 11516 6t3e18 11518 6t4e24 11519 6t5e30 11520 6t5e30OLD 11521 6t6e36 11522 6t6e36OLD 11523 7t3e21 11525 7t4e28 11526 7t5e35 11527 7t6e42 11528 7t7e49 11529 8t3e24 11531 8t4e32 11532 8t5e40 11533 8t5e40OLD 11534 8t6e48 11535 8t6e48OLD 11536 8t7e56 11537 8t8e64 11538 9t3e27 11540 9t4e36 11541 9t5e45 11542 9t6e54 11543 9t7e63 11544 9t8e72 11545 9t9e81 11546 |
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