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| Mirrors > Home > MPE Home > Th. List > zlmlemOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of zlmlem 21527 as of 3-Nov-2024. Lemma for zlmbas 21529 and zlmplusg 21531. (Contributed by Mario Carneiro, 2-Oct-2015.) (New usage is discouraged.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| zlmbas.w | ⊢ 𝑊 = (ℤMod‘𝐺) |
| zlmlemOLD.2 | ⊢ 𝐸 = Slot 𝑁 |
| zlmlemOLD.3 | ⊢ 𝑁 ∈ ℕ |
| zlmlemOLD.4 | ⊢ 𝑁 < 5 |
| Ref | Expression |
|---|---|
| zlmlemOLD | ⊢ (𝐸‘𝐺) = (𝐸‘𝑊) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | zlmlemOLD.2 | . . . . . 6 ⊢ 𝐸 = Slot 𝑁 | |
| 2 | zlmlemOLD.3 | . . . . . 6 ⊢ 𝑁 ∈ ℕ | |
| 3 | 1, 2 | ndxid 17234 | . . . . 5 ⊢ 𝐸 = Slot (𝐸‘ndx) |
| 4 | 1, 2 | ndxarg 17233 | . . . . . . . 8 ⊢ (𝐸‘ndx) = 𝑁 |
| 5 | 2 | nnrei 12275 | . . . . . . . 8 ⊢ 𝑁 ∈ ℝ |
| 6 | 4, 5 | eqeltri 2837 | . . . . . . 7 ⊢ (𝐸‘ndx) ∈ ℝ |
| 7 | zlmlemOLD.4 | . . . . . . . 8 ⊢ 𝑁 < 5 | |
| 8 | 4, 7 | eqbrtri 5164 | . . . . . . 7 ⊢ (𝐸‘ndx) < 5 |
| 9 | 6, 8 | ltneii 11374 | . . . . . 6 ⊢ (𝐸‘ndx) ≠ 5 |
| 10 | scandx 17358 | . . . . . 6 ⊢ (Scalar‘ndx) = 5 | |
| 11 | 9, 10 | neeqtrri 3014 | . . . . 5 ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx) |
| 12 | 3, 11 | setsnid 17245 | . . . 4 ⊢ (𝐸‘𝐺) = (𝐸‘(𝐺 sSet 〈(Scalar‘ndx), ℤring〉)) |
| 13 | 5lt6 12447 | . . . . . . . 8 ⊢ 5 < 6 | |
| 14 | 5re 12353 | . . . . . . . . 9 ⊢ 5 ∈ ℝ | |
| 15 | 6re 12356 | . . . . . . . . 9 ⊢ 6 ∈ ℝ | |
| 16 | 6, 14, 15 | lttri 11387 | . . . . . . . 8 ⊢ (((𝐸‘ndx) < 5 ∧ 5 < 6) → (𝐸‘ndx) < 6) |
| 17 | 8, 13, 16 | mp2an 692 | . . . . . . 7 ⊢ (𝐸‘ndx) < 6 |
| 18 | 6, 17 | ltneii 11374 | . . . . . 6 ⊢ (𝐸‘ndx) ≠ 6 |
| 19 | vscandx 17363 | . . . . . 6 ⊢ ( ·𝑠 ‘ndx) = 6 | |
| 20 | 18, 19 | neeqtrri 3014 | . . . . 5 ⊢ (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx) |
| 21 | 3, 20 | setsnid 17245 | . . . 4 ⊢ (𝐸‘(𝐺 sSet 〈(Scalar‘ndx), ℤring〉)) = (𝐸‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
| 22 | 12, 21 | eqtri 2765 | . . 3 ⊢ (𝐸‘𝐺) = (𝐸‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
| 23 | zlmbas.w | . . . . 5 ⊢ 𝑊 = (ℤMod‘𝐺) | |
| 24 | eqid 2737 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
| 25 | 23, 24 | zlmval 21526 | . . . 4 ⊢ (𝐺 ∈ V → 𝑊 = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
| 26 | 25 | fveq2d 6910 | . . 3 ⊢ (𝐺 ∈ V → (𝐸‘𝑊) = (𝐸‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉))) |
| 27 | 22, 26 | eqtr4id 2796 | . 2 ⊢ (𝐺 ∈ V → (𝐸‘𝐺) = (𝐸‘𝑊)) |
| 28 | 1 | str0 17226 | . . 3 ⊢ ∅ = (𝐸‘∅) |
| 29 | fvprc 6898 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝐸‘𝐺) = ∅) | |
| 30 | fvprc 6898 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → (ℤMod‘𝐺) = ∅) | |
| 31 | 23, 30 | eqtrid 2789 | . . . 4 ⊢ (¬ 𝐺 ∈ V → 𝑊 = ∅) |
| 32 | 31 | fveq2d 6910 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝐸‘𝑊) = (𝐸‘∅)) |
| 33 | 28, 29, 32 | 3eqtr4a 2803 | . 2 ⊢ (¬ 𝐺 ∈ V → (𝐸‘𝐺) = (𝐸‘𝑊)) |
| 34 | 27, 33 | pm2.61i 182 | 1 ⊢ (𝐸‘𝐺) = (𝐸‘𝑊) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 〈cop 4632 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 < clt 11295 ℕcn 12266 5c5 12324 6c6 12325 sSet csts 17200 Slot cslot 17218 ndxcnx 17230 Scalarcsca 17300 ·𝑠 cvsca 17301 .gcmg 19085 ℤringczring 21457 ℤModczlm 21511 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-sets 17201 df-slot 17219 df-ndx 17231 df-sca 17313 df-vsca 17314 df-zlm 21515 |
| This theorem is referenced by: zlmbasOLD 21530 zlmplusgOLD 21532 zlmmulrOLD 21534 |
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