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Mirrors > Home > MPE Home > Th. List > zlmlem | Structured version Visualization version GIF version |
Description: Lemma for zlmbas 20211 and zlmplusg 20212. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
zlmbas.w | ⊢ 𝑊 = (ℤMod‘𝐺) |
zlmlem.2 | ⊢ 𝐸 = Slot 𝑁 |
zlmlem.3 | ⊢ 𝑁 ∈ ℕ |
zlmlem.4 | ⊢ 𝑁 < 5 |
Ref | Expression |
---|---|
zlmlem | ⊢ (𝐸‘𝐺) = (𝐸‘𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zlmlem.2 | . . . . . 6 ⊢ 𝐸 = Slot 𝑁 | |
2 | zlmlem.3 | . . . . . 6 ⊢ 𝑁 ∈ ℕ | |
3 | 1, 2 | ndxid 16501 | . . . . 5 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | 1, 2 | ndxarg 16500 | . . . . . . . 8 ⊢ (𝐸‘ndx) = 𝑁 |
5 | 2 | nnrei 11634 | . . . . . . . 8 ⊢ 𝑁 ∈ ℝ |
6 | 4, 5 | eqeltri 2886 | . . . . . . 7 ⊢ (𝐸‘ndx) ∈ ℝ |
7 | zlmlem.4 | . . . . . . . 8 ⊢ 𝑁 < 5 | |
8 | 4, 7 | eqbrtri 5051 | . . . . . . 7 ⊢ (𝐸‘ndx) < 5 |
9 | 6, 8 | ltneii 10742 | . . . . . 6 ⊢ (𝐸‘ndx) ≠ 5 |
10 | scandx 16624 | . . . . . 6 ⊢ (Scalar‘ndx) = 5 | |
11 | 9, 10 | neeqtrri 3060 | . . . . 5 ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx) |
12 | 3, 11 | setsnid 16531 | . . . 4 ⊢ (𝐸‘𝐺) = (𝐸‘(𝐺 sSet 〈(Scalar‘ndx), ℤring〉)) |
13 | 5lt6 11806 | . . . . . . . 8 ⊢ 5 < 6 | |
14 | 5re 11712 | . . . . . . . . 9 ⊢ 5 ∈ ℝ | |
15 | 6re 11715 | . . . . . . . . 9 ⊢ 6 ∈ ℝ | |
16 | 6, 14, 15 | lttri 10755 | . . . . . . . 8 ⊢ (((𝐸‘ndx) < 5 ∧ 5 < 6) → (𝐸‘ndx) < 6) |
17 | 8, 13, 16 | mp2an 691 | . . . . . . 7 ⊢ (𝐸‘ndx) < 6 |
18 | 6, 17 | ltneii 10742 | . . . . . 6 ⊢ (𝐸‘ndx) ≠ 6 |
19 | vscandx 16626 | . . . . . 6 ⊢ ( ·𝑠 ‘ndx) = 6 | |
20 | 18, 19 | neeqtrri 3060 | . . . . 5 ⊢ (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx) |
21 | 3, 20 | setsnid 16531 | . . . 4 ⊢ (𝐸‘(𝐺 sSet 〈(Scalar‘ndx), ℤring〉)) = (𝐸‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
22 | 12, 21 | eqtri 2821 | . . 3 ⊢ (𝐸‘𝐺) = (𝐸‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
23 | zlmbas.w | . . . . 5 ⊢ 𝑊 = (ℤMod‘𝐺) | |
24 | eqid 2798 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
25 | 23, 24 | zlmval 20209 | . . . 4 ⊢ (𝐺 ∈ V → 𝑊 = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
26 | 25 | fveq2d 6649 | . . 3 ⊢ (𝐺 ∈ V → (𝐸‘𝑊) = (𝐸‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉))) |
27 | 22, 26 | eqtr4id 2852 | . 2 ⊢ (𝐺 ∈ V → (𝐸‘𝐺) = (𝐸‘𝑊)) |
28 | 1 | str0 16527 | . . 3 ⊢ ∅ = (𝐸‘∅) |
29 | fvprc 6638 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝐸‘𝐺) = ∅) | |
30 | fvprc 6638 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → (ℤMod‘𝐺) = ∅) | |
31 | 23, 30 | syl5eq 2845 | . . . 4 ⊢ (¬ 𝐺 ∈ V → 𝑊 = ∅) |
32 | 31 | fveq2d 6649 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝐸‘𝑊) = (𝐸‘∅)) |
33 | 28, 29, 32 | 3eqtr4a 2859 | . 2 ⊢ (¬ 𝐺 ∈ V → (𝐸‘𝐺) = (𝐸‘𝑊)) |
34 | 27, 33 | pm2.61i 185 | 1 ⊢ (𝐸‘𝐺) = (𝐸‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 〈cop 4531 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 < clt 10664 ℕcn 11625 5c5 11683 6c6 11684 ndxcnx 16472 sSet csts 16473 Slot cslot 16474 Scalarcsca 16560 ·𝑠 cvsca 16561 .gcmg 18216 ℤringzring 20163 ℤModczlm 20194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-ndx 16478 df-slot 16479 df-sets 16482 df-sca 16573 df-vsca 16574 df-zlm 20198 |
This theorem is referenced by: zlmbas 20211 zlmplusg 20212 zlmmulr 20213 |
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