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Mirrors > Home > MPE Home > Th. List > zlmlem | Structured version Visualization version GIF version |
Description: Lemma for zlmbas 20942 and zlmplusg 20944. (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 17089 | . . . 4 ⊢ (𝐸‘𝐺) = (𝐸‘(𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩)) |
4 | zlmlem.4 | . . . . 5 ⊢ (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx) | |
5 | 1, 4 | setsnid 17089 | . . . 4 ⊢ (𝐸‘(𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩)) = (𝐸‘((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝐺)⟩)) |
6 | 3, 5 | eqtri 2761 | . . 3 ⊢ (𝐸‘𝐺) = (𝐸‘((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝐺)⟩)) |
7 | zlmbas.w | . . . . 5 ⊢ 𝑊 = (ℤMod‘𝐺) | |
8 | eqid 2733 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
9 | 7, 8 | zlmval 20939 | . . . 4 ⊢ (𝐺 ∈ V → 𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝐺)⟩)) |
10 | 9 | fveq2d 6850 | . . 3 ⊢ (𝐺 ∈ V → (𝐸‘𝑊) = (𝐸‘((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝐺)⟩))) |
11 | 6, 10 | eqtr4id 2792 | . 2 ⊢ (𝐺 ∈ V → (𝐸‘𝐺) = (𝐸‘𝑊)) |
12 | 1 | str0 17069 | . . . 4 ⊢ ∅ = (𝐸‘∅) |
13 | 12 | eqcomi 2742 | . . 3 ⊢ (𝐸‘∅) = ∅ |
14 | 13, 7 | fveqprc 17071 | . 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 1542 ∈ wcel 2107 ≠ wne 2940 Vcvv 3447 ∅c0 4286 ⟨cop 4596 ‘cfv 6500 (class class class)co 7361 sSet csts 17043 Slot cslot 17061 ndxcnx 17073 Scalarcsca 17144 ·𝑠 cvsca 17145 .gcmg 18880 ℤringczring 20892 ℤModczlm 20924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-res 5649 df-iota 6452 df-fun 6502 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-sets 17044 df-slot 17062 df-zlm 20928 |
This theorem is referenced by: zlmbas 20942 zlmplusg 20944 zlmmulr 20946 |
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