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Mirrors > Home > MPE Home > Th. List > zlmlem | Structured version Visualization version GIF version |
Description: Lemma for zlmbas 20668 and zlmplusg 20669. (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 | zlmbas.w | . . . . 5 ⊢ 𝑊 = (ℤMod‘𝐺) | |
2 | eqid 2824 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
3 | 1, 2 | zlmval 20666 | . . . 4 ⊢ (𝐺 ∈ V → 𝑊 = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
4 | 3 | fveq2d 6677 | . . 3 ⊢ (𝐺 ∈ V → (𝐸‘𝑊) = (𝐸‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉))) |
5 | zlmlem.2 | . . . . . 6 ⊢ 𝐸 = Slot 𝑁 | |
6 | zlmlem.3 | . . . . . 6 ⊢ 𝑁 ∈ ℕ | |
7 | 5, 6 | ndxid 16512 | . . . . 5 ⊢ 𝐸 = Slot (𝐸‘ndx) |
8 | 5, 6 | ndxarg 16511 | . . . . . . . 8 ⊢ (𝐸‘ndx) = 𝑁 |
9 | 6 | nnrei 11650 | . . . . . . . 8 ⊢ 𝑁 ∈ ℝ |
10 | 8, 9 | eqeltri 2912 | . . . . . . 7 ⊢ (𝐸‘ndx) ∈ ℝ |
11 | zlmlem.4 | . . . . . . . 8 ⊢ 𝑁 < 5 | |
12 | 8, 11 | eqbrtri 5090 | . . . . . . 7 ⊢ (𝐸‘ndx) < 5 |
13 | 10, 12 | ltneii 10756 | . . . . . 6 ⊢ (𝐸‘ndx) ≠ 5 |
14 | scandx 16635 | . . . . . 6 ⊢ (Scalar‘ndx) = 5 | |
15 | 13, 14 | neeqtrri 3092 | . . . . 5 ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx) |
16 | 7, 15 | setsnid 16542 | . . . 4 ⊢ (𝐸‘𝐺) = (𝐸‘(𝐺 sSet 〈(Scalar‘ndx), ℤring〉)) |
17 | 5lt6 11821 | . . . . . . . 8 ⊢ 5 < 6 | |
18 | 5re 11727 | . . . . . . . . 9 ⊢ 5 ∈ ℝ | |
19 | 6re 11730 | . . . . . . . . 9 ⊢ 6 ∈ ℝ | |
20 | 10, 18, 19 | lttri 10769 | . . . . . . . 8 ⊢ (((𝐸‘ndx) < 5 ∧ 5 < 6) → (𝐸‘ndx) < 6) |
21 | 12, 17, 20 | mp2an 690 | . . . . . . 7 ⊢ (𝐸‘ndx) < 6 |
22 | 10, 21 | ltneii 10756 | . . . . . 6 ⊢ (𝐸‘ndx) ≠ 6 |
23 | vscandx 16637 | . . . . . 6 ⊢ ( ·𝑠 ‘ndx) = 6 | |
24 | 22, 23 | neeqtrri 3092 | . . . . 5 ⊢ (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx) |
25 | 7, 24 | setsnid 16542 | . . . 4 ⊢ (𝐸‘(𝐺 sSet 〈(Scalar‘ndx), ℤring〉)) = (𝐸‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
26 | 16, 25 | eqtri 2847 | . . 3 ⊢ (𝐸‘𝐺) = (𝐸‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
27 | 4, 26 | syl6reqr 2878 | . 2 ⊢ (𝐺 ∈ V → (𝐸‘𝐺) = (𝐸‘𝑊)) |
28 | 5 | str0 16538 | . . 3 ⊢ ∅ = (𝐸‘∅) |
29 | fvprc 6666 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝐸‘𝐺) = ∅) | |
30 | fvprc 6666 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → (ℤMod‘𝐺) = ∅) | |
31 | 1, 30 | syl5eq 2871 | . . . 4 ⊢ (¬ 𝐺 ∈ V → 𝑊 = ∅) |
32 | 31 | fveq2d 6677 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝐸‘𝑊) = (𝐸‘∅)) |
33 | 28, 29, 32 | 3eqtr4a 2885 | . 2 ⊢ (¬ 𝐺 ∈ V → (𝐸‘𝐺) = (𝐸‘𝑊)) |
34 | 27, 33 | pm2.61i 184 | 1 ⊢ (𝐸‘𝐺) = (𝐸‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∅c0 4294 〈cop 4576 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 ℝcr 10539 < clt 10678 ℕcn 11641 5c5 11698 6c6 11699 ndxcnx 16483 sSet csts 16484 Slot cslot 16485 Scalarcsca 16571 ·𝑠 cvsca 16572 .gcmg 18227 ℤringzring 20620 ℤModczlm 20651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-ndx 16489 df-slot 16490 df-sets 16493 df-sca 16584 df-vsca 16585 df-zlm 20655 |
This theorem is referenced by: zlmbas 20668 zlmplusg 20669 zlmmulr 20670 |
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