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| Mirrors > Home > MPE Home > Th. List > gexid | Structured version Visualization version GIF version | ||
| Description: Any element to the power of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.) |
| Ref | Expression |
|---|---|
| gexcl.1 | ⊢ 𝑋 = (Base‘𝐺) |
| gexcl.2 | ⊢ 𝐸 = (gEx‘𝐺) |
| gexid.3 | ⊢ · = (.g‘𝐺) |
| gexid.4 | ⊢ 0 = (0g‘𝐺) |
| Ref | Expression |
|---|---|
| gexid | ⊢ (𝐴 ∈ 𝑋 → (𝐸 · 𝐴) = 0 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | oveq1 7438 | . . . 4 ⊢ (𝐸 = 0 → (𝐸 · 𝐴) = (0 · 𝐴)) | |
| 2 | gexcl.1 | . . . . 5 ⊢ 𝑋 = (Base‘𝐺) | |
| 3 | gexid.4 | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
| 4 | gexid.3 | . . . . 5 ⊢ · = (.g‘𝐺) | |
| 5 | 2, 3, 4 | mulg0 19092 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → (0 · 𝐴) = 0 ) |
| 6 | 1, 5 | sylan9eqr 2799 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐸 = 0) → (𝐸 · 𝐴) = 0 ) |
| 7 | 6 | adantrr 717 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ (𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅)) → (𝐸 · 𝐴) = 0 ) |
| 8 | oveq1 7438 | . . . . . . 7 ⊢ (𝑦 = 𝐸 → (𝑦 · 𝑥) = (𝐸 · 𝑥)) | |
| 9 | 8 | eqeq1d 2739 | . . . . . 6 ⊢ (𝑦 = 𝐸 → ((𝑦 · 𝑥) = 0 ↔ (𝐸 · 𝑥) = 0 )) |
| 10 | 9 | ralbidv 3178 | . . . . 5 ⊢ (𝑦 = 𝐸 → (∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 ↔ ∀𝑥 ∈ 𝑋 (𝐸 · 𝑥) = 0 )) |
| 11 | 10 | elrab 3692 | . . . 4 ⊢ (𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } ↔ (𝐸 ∈ ℕ ∧ ∀𝑥 ∈ 𝑋 (𝐸 · 𝑥) = 0 )) |
| 12 | 11 | simprbi 496 | . . 3 ⊢ (𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } → ∀𝑥 ∈ 𝑋 (𝐸 · 𝑥) = 0 ) |
| 13 | oveq2 7439 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐸 · 𝑥) = (𝐸 · 𝐴)) | |
| 14 | 13 | eqeq1d 2739 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝐸 · 𝑥) = 0 ↔ (𝐸 · 𝐴) = 0 )) |
| 15 | 14 | rspcva 3620 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝐸 · 𝑥) = 0 ) → (𝐸 · 𝐴) = 0 ) |
| 16 | 12, 15 | sylan2 593 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 }) → (𝐸 · 𝐴) = 0 ) |
| 17 | elfvex 6944 | . . . 4 ⊢ (𝐴 ∈ (Base‘𝐺) → 𝐺 ∈ V) | |
| 18 | 17, 2 | eleq2s 2859 | . . 3 ⊢ (𝐴 ∈ 𝑋 → 𝐺 ∈ V) |
| 19 | gexcl.2 | . . . 4 ⊢ 𝐸 = (gEx‘𝐺) | |
| 20 | eqid 2737 | . . . 4 ⊢ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } | |
| 21 | 2, 4, 3, 19, 20 | gexlem1 19597 | . . 3 ⊢ (𝐺 ∈ V → ((𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅) ∨ 𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 })) |
| 22 | 18, 21 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑋 → ((𝐸 = 0 ∧ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 } = ∅) ∨ 𝐸 ∈ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 })) |
| 23 | 7, 16, 22 | mpjaodan 961 | 1 ⊢ (𝐴 ∈ 𝑋 → (𝐸 · 𝐴) = 0 ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 Vcvv 3480 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 0cc0 11155 ℕcn 12266 Basecbs 17247 0gc0g 17484 .gcmg 19085 gExcgex 19543 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-seq 14043 df-mulg 19086 df-gex 19547 |
| This theorem is referenced by: gexdvdsi 19601 gexod 19604 gex1 19609 pgpfac1lem3a 20096 |
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