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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 6886 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 · 𝑥) = (𝑦 · 𝐴)) | |
2 | 1 | eqeq1d 2801 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑦 · 𝑥) = 0 ↔ (𝑦 · 𝐴) = 0 )) |
3 | 2 | rabbidv 3373 | . . . . 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 2851 | . . . 4 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = 𝐼) |
6 | 5 | csbeq1d 3735 | . . 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 11319 | . . . . 5 ⊢ ℕ ∈ V | |
8 | 4, 7 | rabex2 5009 | . . . 4 ⊢ 𝐼 ∈ V |
9 | eqeq1 2803 | . . . . 5 ⊢ (𝑖 = 𝐼 → (𝑖 = ∅ ↔ 𝐼 = ∅)) | |
10 | infeq1 8624 | . . . . 5 ⊢ (𝑖 = 𝐼 → inf(𝑖, ℝ, < ) = inf(𝐼, ℝ, < )) | |
11 | 9, 10 | ifbieq2d 4302 | . . . 4 ⊢ (𝑖 = 𝐼 → if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < ))) |
12 | 8, 11 | csbie 3754 | . . 3 ⊢ ⦋𝐼 / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) |
13 | 6, 12 | syl6eq 2849 | . 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 18265 | . 2 ⊢ 𝑂 = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) |
19 | c0ex 10322 | . . 3 ⊢ 0 ∈ V | |
20 | ltso 10408 | . . . 4 ⊢ < Or ℝ | |
21 | 20 | infex 8641 | . . 3 ⊢ inf(𝐼, ℝ, < ) ∈ V |
22 | 19, 21 | ifex 4325 | . 2 ⊢ if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) ∈ V |
23 | 13, 18, 22 | fvmpt 6507 | 1 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 {crab 3093 ⦋csb 3728 ∅c0 4115 ifcif 4277 ‘cfv 6101 (class class class)co 6878 infcinf 8589 ℝcr 10223 0cc0 10224 < clt 10363 ℕcn 11312 Basecbs 16184 0gc0g 16415 .gcmg 17856 odcod 18257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-mulcl 10286 ax-i2m1 10292 ax-pre-lttri 10298 ax-pre-lttrn 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-om 7300 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-sup 8590 df-inf 8591 df-pnf 10365 df-mnf 10366 df-ltxr 10368 df-nn 11313 df-od 18261 |
This theorem is referenced by: odlem1 18267 odlem2 18271 submod 18297 ofldchr 30330 |
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