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Mirrors > Home > MPE Home > Th. List > 4t3lem | Structured version Visualization version GIF version |
Description: Lemma for 4t3e12 12199 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 7170 | . 2 ⊢ (𝐴 · 𝐶) = (𝐴 · (𝐵 + 1)) |
3 | 4t3lem.1 | . . . . . 6 ⊢ 𝐴 ∈ ℕ0 | |
4 | 3 | nn0cni 11912 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
5 | 4t3lem.2 | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
6 | 5 | nn0cni 11912 | . . . . 5 ⊢ 𝐵 ∈ ℂ |
7 | ax-1cn 10598 | . . . . 5 ⊢ 1 ∈ ℂ | |
8 | 4, 6, 7 | adddii 10656 | . . . 4 ⊢ (𝐴 · (𝐵 + 1)) = ((𝐴 · 𝐵) + (𝐴 · 1)) |
9 | 4t3lem.4 | . . . . 5 ⊢ (𝐴 · 𝐵) = 𝐷 | |
10 | 4 | mulid1i 10648 | . . . . 5 ⊢ (𝐴 · 1) = 𝐴 |
11 | 9, 10 | oveq12i 7171 | . . . 4 ⊢ ((𝐴 · 𝐵) + (𝐴 · 1)) = (𝐷 + 𝐴) |
12 | 8, 11 | eqtri 2847 | . . 3 ⊢ (𝐴 · (𝐵 + 1)) = (𝐷 + 𝐴) |
13 | 4t3lem.5 | . . 3 ⊢ (𝐷 + 𝐴) = 𝐸 | |
14 | 12, 13 | eqtri 2847 | . 2 ⊢ (𝐴 · (𝐵 + 1)) = 𝐸 |
15 | 2, 14 | eqtri 2847 | 1 ⊢ (𝐴 · 𝐶) = 𝐸 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 (class class class)co 7159 1c1 10541 + caddc 10543 · cmul 10545 ℕ0cn0 11900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-mulcl 10602 ax-mulcom 10604 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1rid 10610 ax-cnre 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-nn 11642 df-n0 11901 |
This theorem is referenced by: 4t3e12 12199 4t4e16 12200 5t2e10 12201 5t3e15 12202 5t4e20 12203 5t5e25 12204 6t3e18 12206 6t4e24 12207 6t5e30 12208 6t6e36 12209 7t3e21 12211 7t4e28 12212 7t5e35 12213 7t6e42 12214 7t7e49 12215 8t3e24 12217 8t4e32 12218 8t5e40 12219 8t6e48 12220 8t7e56 12221 8t8e64 12222 9t3e27 12224 9t4e36 12225 9t5e45 12226 9t6e54 12227 9t7e63 12228 9t8e72 12229 9t9e81 12230 |
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