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Mirrors > Home > MPE Home > Th. List > zlmlem | Structured version Visualization version GIF version |
Description: Lemma for zlmbas 21554 and zlmplusg 21556. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.) |
Ref | Expression |
---|---|
zlmbas.w | ⊢ 𝑊 = (ℤMod‘𝐺) |
zlmlem.2 | ⊢ 𝐸 = Slot (𝐸‘ndx) |
zlmlem.3 | ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx) |
zlmlem.4 | ⊢ (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx) |
Ref | Expression |
---|---|
zlmlem | ⊢ (𝐸‘𝐺) = (𝐸‘𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zlmlem.2 | . . . . 5 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
2 | zlmlem.3 | . . . . 5 ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx) | |
3 | 1, 2 | setsnid 17258 | . . . 4 ⊢ (𝐸‘𝐺) = (𝐸‘(𝐺 sSet 〈(Scalar‘ndx), ℤring〉)) |
4 | zlmlem.4 | . . . . 5 ⊢ (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx) | |
5 | 1, 4 | setsnid 17258 | . . . 4 ⊢ (𝐸‘(𝐺 sSet 〈(Scalar‘ndx), ℤring〉)) = (𝐸‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
6 | 3, 5 | eqtri 2768 | . . 3 ⊢ (𝐸‘𝐺) = (𝐸‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
7 | zlmbas.w | . . . . 5 ⊢ 𝑊 = (ℤMod‘𝐺) | |
8 | eqid 2740 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
9 | 7, 8 | zlmval 21551 | . . . 4 ⊢ (𝐺 ∈ V → 𝑊 = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
10 | 9 | fveq2d 6926 | . . 3 ⊢ (𝐺 ∈ V → (𝐸‘𝑊) = (𝐸‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉))) |
11 | 6, 10 | eqtr4id 2799 | . 2 ⊢ (𝐺 ∈ V → (𝐸‘𝐺) = (𝐸‘𝑊)) |
12 | 1 | str0 17238 | . . . 4 ⊢ ∅ = (𝐸‘∅) |
13 | 12 | eqcomi 2749 | . . 3 ⊢ (𝐸‘∅) = ∅ |
14 | 13, 7 | fveqprc 17240 | . 2 ⊢ (¬ 𝐺 ∈ V → (𝐸‘𝐺) = (𝐸‘𝑊)) |
15 | 11, 14 | pm2.61i 182 | 1 ⊢ (𝐸‘𝐺) = (𝐸‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2108 ≠ wne 2946 Vcvv 3488 ∅c0 4352 〈cop 4654 ‘cfv 6575 (class class class)co 7450 sSet csts 17212 Slot cslot 17230 ndxcnx 17242 Scalarcsca 17316 ·𝑠 cvsca 17317 .gcmg 19109 ℤringczring 21482 ℤModczlm 21536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7772 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-res 5712 df-iota 6527 df-fun 6577 df-fv 6583 df-ov 7453 df-oprab 7454 df-mpo 7455 df-sets 17213 df-slot 17231 df-zlm 21540 |
This theorem is referenced by: zlmbas 21554 zlmplusg 21556 zlmmulr 21558 |
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