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Mirrors > Home > MPE Home > Th. List > mulgnnp1 | Structured version Visualization version GIF version |
Description: Group multiple (exponentiation) operation at a successor. (Contributed by Mario Carneiro, 11-Dec-2014.) |
Ref | Expression |
---|---|
mulg1.b | ⊢ 𝐵 = (Base‘𝐺) |
mulg1.m | ⊢ · = (.g‘𝐺) |
mulgnnp1.p | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
mulgnnp1 | ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 481 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → 𝑁 ∈ ℕ) | |
2 | nnuz 12898 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
3 | 1, 2 | eleqtrdi 2835 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → 𝑁 ∈ (ℤ≥‘1)) |
4 | seqp1 14017 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘1) → (seq1( + , (ℕ × {𝑋}))‘(𝑁 + 1)) = ((seq1( + , (ℕ × {𝑋}))‘𝑁) + ((ℕ × {𝑋})‘(𝑁 + 1)))) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (seq1( + , (ℕ × {𝑋}))‘(𝑁 + 1)) = ((seq1( + , (ℕ × {𝑋}))‘𝑁) + ((ℕ × {𝑋})‘(𝑁 + 1)))) |
6 | id 22 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐵) | |
7 | peano2nn 12257 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
8 | fvconst2g 7214 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑁 + 1) ∈ ℕ) → ((ℕ × {𝑋})‘(𝑁 + 1)) = 𝑋) | |
9 | 6, 7, 8 | syl2anr 595 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((ℕ × {𝑋})‘(𝑁 + 1)) = 𝑋) |
10 | 9 | oveq2d 7435 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((seq1( + , (ℕ × {𝑋}))‘𝑁) + ((ℕ × {𝑋})‘(𝑁 + 1))) = ((seq1( + , (ℕ × {𝑋}))‘𝑁) + 𝑋)) |
11 | 5, 10 | eqtrd 2765 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (seq1( + , (ℕ × {𝑋}))‘(𝑁 + 1)) = ((seq1( + , (ℕ × {𝑋}))‘𝑁) + 𝑋)) |
12 | mulg1.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
13 | mulgnnp1.p | . . . 4 ⊢ + = (+g‘𝐺) | |
14 | mulg1.m | . . . 4 ⊢ · = (.g‘𝐺) | |
15 | eqid 2725 | . . . 4 ⊢ seq1( + , (ℕ × {𝑋})) = seq1( + , (ℕ × {𝑋})) | |
16 | 12, 13, 14, 15 | mulgnn 19039 | . . 3 ⊢ (((𝑁 + 1) ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = (seq1( + , (ℕ × {𝑋}))‘(𝑁 + 1))) |
17 | 7, 16 | sylan 578 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = (seq1( + , (ℕ × {𝑋}))‘(𝑁 + 1))) |
18 | 12, 13, 14, 15 | mulgnn 19039 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (seq1( + , (ℕ × {𝑋}))‘𝑁)) |
19 | 18 | oveq1d 7434 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((𝑁 · 𝑋) + 𝑋) = ((seq1( + , (ℕ × {𝑋}))‘𝑁) + 𝑋)) |
20 | 11, 17, 19 | 3eqtr4d 2775 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {csn 4630 × cxp 5676 ‘cfv 6549 (class class class)co 7419 1c1 11141 + caddc 11143 ℕcn 12245 ℤ≥cuz 12855 seqcseq 14002 Basecbs 17183 +gcplusg 17236 .gcmg 19031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-n0 12506 df-z 12592 df-uz 12856 df-seq 14003 df-mulg 19032 |
This theorem is referenced by: mulg2 19046 mulgnn0p1 19048 mulgnnass 19072 chfacfpmmulgsum2 22811 xrsmulgzz 32825 ofldchr 33128 ringexp0nn 41737 |
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