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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 483 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → 𝑁 ∈ ℕ) | |
2 | nnuz 12631 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
3 | 1, 2 | eleqtrdi 2849 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → 𝑁 ∈ (ℤ≥‘1)) |
4 | seqp1 13746 | . . . 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 11995 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
8 | fvconst2g 7069 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑁 + 1) ∈ ℕ) → ((ℕ × {𝑋})‘(𝑁 + 1)) = 𝑋) | |
9 | 6, 7, 8 | syl2anr 597 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((ℕ × {𝑋})‘(𝑁 + 1)) = 𝑋) |
10 | 9 | oveq2d 7283 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((seq1( + , (ℕ × {𝑋}))‘𝑁) + ((ℕ × {𝑋})‘(𝑁 + 1))) = ((seq1( + , (ℕ × {𝑋}))‘𝑁) + 𝑋)) |
11 | 5, 10 | eqtrd 2778 | . 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 2738 | . . . 4 ⊢ seq1( + , (ℕ × {𝑋})) = seq1( + , (ℕ × {𝑋})) | |
16 | 12, 13, 14, 15 | mulgnn 18718 | . . 3 ⊢ (((𝑁 + 1) ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = (seq1( + , (ℕ × {𝑋}))‘(𝑁 + 1))) |
17 | 7, 16 | sylan 580 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = (seq1( + , (ℕ × {𝑋}))‘(𝑁 + 1))) |
18 | 12, 13, 14, 15 | mulgnn 18718 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (seq1( + , (ℕ × {𝑋}))‘𝑁)) |
19 | 18 | oveq1d 7282 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((𝑁 · 𝑋) + 𝑋) = ((seq1( + , (ℕ × {𝑋}))‘𝑁) + 𝑋)) |
20 | 11, 17, 19 | 3eqtr4d 2788 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {csn 4561 × cxp 5582 ‘cfv 6426 (class class class)co 7267 1c1 10882 + caddc 10884 ℕcn 11983 ℤ≥cuz 12592 seqcseq 13731 Basecbs 16922 +gcplusg 16972 .gcmg 18710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-n0 12244 df-z 12330 df-uz 12593 df-seq 13732 df-mulg 18711 |
This theorem is referenced by: mulg2 18723 mulgnn0p1 18725 mulgnnass 18748 chfacfpmmulgsum2 22024 xrsmulgzz 31295 ofldchr 31521 |
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