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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 6557 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 · 𝑥) = (𝑦 · 𝐴)) | |
2 | 1 | eqeq1d 2612 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑦 · 𝑥) = 0 ↔ (𝑦 · 𝐴) = 0 )) |
3 | 2 | rabbidv 3164 | . . . . 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 2662 | . . . 4 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } = 𝐼) |
6 | 5 | csbeq1d 3506 | . . 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 10903 | . . . . 5 ⊢ ℕ ∈ V | |
8 | 4, 7 | rabex2 4742 | . . . 4 ⊢ 𝐼 ∈ V |
9 | eqeq1 2614 | . . . . 5 ⊢ (𝑖 = 𝐼 → (𝑖 = ∅ ↔ 𝐼 = ∅)) | |
10 | infeq1 8265 | . . . . 5 ⊢ (𝑖 = 𝐼 → inf(𝑖, ℝ, < ) = inf(𝐼, ℝ, < )) | |
11 | 9, 10 | ifbieq2d 4061 | . . . 4 ⊢ (𝑖 = 𝐼 → if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < ))) |
12 | 8, 11 | csbie 3525 | . . 3 ⊢ ⦋𝐼 / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) |
13 | 6, 12 | syl6eq 2660 | . 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 17775 | . 2 ⊢ 𝑂 = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) |
19 | c0ex 9913 | . . 3 ⊢ 0 ∈ V | |
20 | ltso 9997 | . . . 4 ⊢ < Or ℝ | |
21 | 20 | infex 8282 | . . 3 ⊢ inf(𝐼, ℝ, < ) ∈ V |
22 | 19, 21 | ifex 4106 | . 2 ⊢ if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) ∈ V |
23 | 13, 18, 22 | fvmpt 6191 | 1 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {crab 2900 ⦋csb 3499 ∅c0 3874 ifcif 4036 ‘cfv 5804 (class class class)co 6549 infcinf 8230 ℝcr 9814 0cc0 9815 < clt 9953 ℕcn 10897 Basecbs 15695 0gc0g 15923 .gcmg 17363 odcod 17767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-nn 10898 df-od 17771 |
This theorem is referenced by: odlem1 17777 odlem2 17781 submod 17807 ofldchr 29145 |
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