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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 6658 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 · 𝑥) = (𝑦 · 𝐴)) | |
2 | 1 | eqeq1d 2624 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑦 · 𝑥) = 0 ↔ (𝑦 · 𝐴) = 0 )) |
3 | 2 | rabbidv 3189 | . . . . 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 2674 | . . . 4 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = 𝐼) |
6 | 5 | csbeq1d 3540 | . . 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 11026 | . . . . 5 ⊢ ℕ ∈ V | |
8 | 4, 7 | rabex2 4815 | . . . 4 ⊢ 𝐼 ∈ V |
9 | eqeq1 2626 | . . . . 5 ⊢ (𝑖 = 𝐼 → (𝑖 = ∅ ↔ 𝐼 = ∅)) | |
10 | infeq1 8382 | . . . . 5 ⊢ (𝑖 = 𝐼 → inf(𝑖, ℝ, < ) = inf(𝐼, ℝ, < )) | |
11 | 9, 10 | ifbieq2d 4111 | . . . 4 ⊢ (𝑖 = 𝐼 → if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < ))) |
12 | 8, 11 | csbie 3559 | . . 3 ⊢ ⦋𝐼 / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) |
13 | 6, 12 | syl6eq 2672 | . 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 17952 | . 2 ⊢ 𝑂 = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) |
19 | c0ex 10034 | . . 3 ⊢ 0 ∈ V | |
20 | ltso 10118 | . . . 4 ⊢ < Or ℝ | |
21 | 20 | infex 8399 | . . 3 ⊢ inf(𝐼, ℝ, < ) ∈ V |
22 | 19, 21 | ifex 4156 | . 2 ⊢ if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) ∈ V |
23 | 13, 18, 22 | fvmpt 6282 | 1 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 {crab 2916 ⦋csb 3533 ∅c0 3915 ifcif 4086 ‘cfv 5888 (class class class)co 6650 infcinf 8347 ℝcr 9935 0cc0 9936 < clt 10074 ℕcn 11020 Basecbs 15857 0gc0g 16100 .gcmg 17540 odcod 17944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-nn 11021 df-od 17948 |
This theorem is referenced by: odlem1 17954 odlem2 17958 submod 17984 ofldchr 29814 |
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