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Mirrors > Home > MPE Home > Th. List > 4t3lem | Structured version Visualization version GIF version |
Description: Lemma for 4t3e12 12190 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 7161 | . 2 ⊢ (𝐴 · 𝐶) = (𝐴 · (𝐵 + 1)) |
3 | 4t3lem.1 | . . . . . 6 ⊢ 𝐴 ∈ ℕ0 | |
4 | 3 | nn0cni 11903 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
5 | 4t3lem.2 | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
6 | 5 | nn0cni 11903 | . . . . 5 ⊢ 𝐵 ∈ ℂ |
7 | ax-1cn 10589 | . . . . 5 ⊢ 1 ∈ ℂ | |
8 | 4, 6, 7 | adddii 10647 | . . . 4 ⊢ (𝐴 · (𝐵 + 1)) = ((𝐴 · 𝐵) + (𝐴 · 1)) |
9 | 4t3lem.4 | . . . . 5 ⊢ (𝐴 · 𝐵) = 𝐷 | |
10 | 4 | mulid1i 10639 | . . . . 5 ⊢ (𝐴 · 1) = 𝐴 |
11 | 9, 10 | oveq12i 7162 | . . . 4 ⊢ ((𝐴 · 𝐵) + (𝐴 · 1)) = (𝐷 + 𝐴) |
12 | 8, 11 | eqtri 2844 | . . 3 ⊢ (𝐴 · (𝐵 + 1)) = (𝐷 + 𝐴) |
13 | 4t3lem.5 | . . 3 ⊢ (𝐷 + 𝐴) = 𝐸 | |
14 | 12, 13 | eqtri 2844 | . 2 ⊢ (𝐴 · (𝐵 + 1)) = 𝐸 |
15 | 2, 14 | eqtri 2844 | 1 ⊢ (𝐴 · 𝐶) = 𝐸 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 (class class class)co 7150 1c1 10532 + caddc 10534 · cmul 10536 ℕ0cn0 11891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-mulcl 10593 ax-mulcom 10595 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1rid 10601 ax-cnre 10604 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-nn 11633 df-n0 11892 |
This theorem is referenced by: 4t3e12 12190 4t4e16 12191 5t2e10 12192 5t3e15 12193 5t4e20 12194 5t5e25 12195 6t3e18 12197 6t4e24 12198 6t5e30 12199 6t6e36 12200 7t3e21 12202 7t4e28 12203 7t5e35 12204 7t6e42 12205 7t7e49 12206 8t3e24 12208 8t4e32 12209 8t5e40 12210 8t6e48 12211 8t7e56 12212 8t8e64 12213 9t3e27 12215 9t4e36 12216 9t5e45 12217 9t6e54 12218 9t7e63 12219 9t8e72 12220 9t9e81 12221 |
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