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Mirrors > Home > MPE Home > Th. List > odval | Structured version Visualization version GIF version |
Description: Second substitution for the group order definition. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 5-Sep-2015.) (Revised by AV, 5-Oct-2020.) |
Ref | Expression |
---|---|
odval.1 | ⊢ 𝑋 = (Base‘𝐺) |
odval.2 | ⊢ · = (.g‘𝐺) |
odval.3 | ⊢ 0 = (0g‘𝐺) |
odval.4 | ⊢ 𝑂 = (od‘𝐺) |
odval.i | ⊢ 𝐼 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 } |
Ref | Expression |
---|---|
odval | ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7163 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 · 𝑥) = (𝑦 · 𝐴)) | |
2 | 1 | eqeq1d 2823 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑦 · 𝑥) = 0 ↔ (𝑦 · 𝐴) = 0 )) |
3 | 2 | rabbidv 3480 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 }) |
4 | odval.i | . . . . 5 ⊢ 𝐼 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 } | |
5 | 3, 4 | syl6eqr 2874 | . . . 4 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = 𝐼) |
6 | 5 | csbeq1d 3886 | . . 3 ⊢ (𝑥 = 𝐴 → ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = ⦋𝐼 / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) |
7 | nnex 11643 | . . . . 5 ⊢ ℕ ∈ V | |
8 | 4, 7 | rabex2 5236 | . . . 4 ⊢ 𝐼 ∈ V |
9 | eqeq1 2825 | . . . . 5 ⊢ (𝑖 = 𝐼 → (𝑖 = ∅ ↔ 𝐼 = ∅)) | |
10 | infeq1 8939 | . . . . 5 ⊢ (𝑖 = 𝐼 → inf(𝑖, ℝ, < ) = inf(𝐼, ℝ, < )) | |
11 | 9, 10 | ifbieq2d 4491 | . . . 4 ⊢ (𝑖 = 𝐼 → if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < ))) |
12 | 8, 11 | csbie 3917 | . . 3 ⊢ ⦋𝐼 / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) |
13 | 6, 12 | syl6eq 2872 | . 2 ⊢ (𝑥 = 𝐴 → ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < ))) |
14 | odval.1 | . . 3 ⊢ 𝑋 = (Base‘𝐺) | |
15 | odval.2 | . . 3 ⊢ · = (.g‘𝐺) | |
16 | odval.3 | . . 3 ⊢ 0 = (0g‘𝐺) | |
17 | odval.4 | . . 3 ⊢ 𝑂 = (od‘𝐺) | |
18 | 14, 15, 16, 17 | odfval 18659 | . 2 ⊢ 𝑂 = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) |
19 | c0ex 10634 | . . 3 ⊢ 0 ∈ V | |
20 | ltso 10720 | . . . 4 ⊢ < Or ℝ | |
21 | 20 | infex 8956 | . . 3 ⊢ inf(𝐼, ℝ, < ) ∈ V |
22 | 19, 21 | ifex 4514 | . 2 ⊢ if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) ∈ V |
23 | 13, 18, 22 | fvmpt 6767 | 1 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {crab 3142 ⦋csb 3882 ∅c0 4290 ifcif 4466 ‘cfv 6354 (class class class)co 7155 infcinf 8904 ℝcr 10535 0cc0 10536 < clt 10674 ℕcn 11637 Basecbs 16482 0gc0g 16712 .gcmg 18223 odcod 18651 |
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 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
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-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 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 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-od 18655 |
This theorem is referenced by: odlem1 18662 odlem2 18666 submod 18693 ofldchr 30887 |
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