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Mirrors > Home > ILE Home > Th. List > mulg1 | GIF version |
Description: Group multiple (exponentiation) operation at one. (Contributed by Mario Carneiro, 11-Dec-2014.) |
Ref | Expression |
---|---|
mulg1.b | ⊢ 𝐵 = (Base‘𝐺) |
mulg1.m | ⊢ · = (.g‘𝐺) |
Ref | Expression |
---|---|
mulg1 | ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn 8906 | . . 3 ⊢ 1 ∈ ℕ | |
2 | mulg1.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | eqid 2177 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | mulg1.m | . . . 4 ⊢ · = (.g‘𝐺) | |
5 | eqid 2177 | . . . 4 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋})) | |
6 | 2, 3, 4, 5 | mulgnn 12865 | . . 3 ⊢ ((1 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (1 · 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘1)) |
7 | 1, 6 | mpan 424 | . 2 ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘1)) |
8 | 1zzd 9256 | . . 3 ⊢ (𝑋 ∈ 𝐵 → 1 ∈ ℤ) | |
9 | elnnuz 9540 | . . . 4 ⊢ (𝑢 ∈ ℕ ↔ 𝑢 ∈ (ℤ≥‘1)) | |
10 | fvconst2g 5725 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑋})‘𝑢) = 𝑋) | |
11 | simpl 109 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → 𝑋 ∈ 𝐵) | |
12 | 10, 11 | eqeltrd 2254 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑋})‘𝑢) ∈ 𝐵) |
13 | 12 | elexd 2750 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑢 ∈ ℕ) → ((ℕ × {𝑋})‘𝑢) ∈ V) |
14 | 9, 13 | sylan2br 288 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑢 ∈ (ℤ≥‘1)) → ((ℕ × {𝑋})‘𝑢) ∈ V) |
15 | simprl 529 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → 𝑢 ∈ V) | |
16 | 2 | basmex 12490 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → 𝐺 ∈ V) |
17 | plusgslid 12538 | . . . . . . 7 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ) | |
18 | 17 | slotex 12459 | . . . . . 6 ⊢ (𝐺 ∈ V → (+g‘𝐺) ∈ V) |
19 | 16, 18 | syl 14 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → (+g‘𝐺) ∈ V) |
20 | 19 | adantr 276 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (+g‘𝐺) ∈ V) |
21 | simprr 531 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → 𝑣 ∈ V) | |
22 | ovexg 5902 | . . . 4 ⊢ ((𝑢 ∈ V ∧ (+g‘𝐺) ∈ V ∧ 𝑣 ∈ V) → (𝑢(+g‘𝐺)𝑣) ∈ V) | |
23 | 15, 20, 21, 22 | syl3anc 1238 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) → (𝑢(+g‘𝐺)𝑣) ∈ V) |
24 | 8, 14, 23 | seq3-1 10433 | . 2 ⊢ (𝑋 ∈ 𝐵 → (seq1((+g‘𝐺), (ℕ × {𝑋}))‘1) = ((ℕ × {𝑋})‘1)) |
25 | fvconst2g 5725 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 1 ∈ ℕ) → ((ℕ × {𝑋})‘1) = 𝑋) | |
26 | 1, 25 | mpan2 425 | . 2 ⊢ (𝑋 ∈ 𝐵 → ((ℕ × {𝑋})‘1) = 𝑋) |
27 | 7, 24, 26 | 3eqtrd 2214 | 1 ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = 𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 Vcvv 2737 {csn 3591 × cxp 4620 ‘cfv 5211 (class class class)co 5868 1c1 7790 ℕcn 8895 ℤ≥cuz 9504 seqcseq 10418 Basecbs 12432 +gcplusg 12505 .gcmg 12859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-ltadd 7905 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4289 df-iord 4362 df-on 4364 df-ilim 4365 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-recs 6299 df-frec 6385 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-inn 8896 df-2 8954 df-n0 9153 df-z 9230 df-uz 9505 df-seqfrec 10419 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-0g 12642 df-minusg 12758 df-mulg 12860 |
This theorem is referenced by: mulg2 12868 mulgnn0p1 12870 mulgm1 12879 mulgp1 12891 mulgnnass 12893 |
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