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Mirrors > Home > MPE Home > Th. List > sralem | Structured version Visualization version GIF version |
Description: Lemma for srabase 20769 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.) |
Ref | Expression |
---|---|
srapart.a | ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
srapart.s | ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) |
sralem.1 | ⊢ 𝐸 = Slot (𝐸‘ndx) |
sralem.2 | ⊢ (Scalar‘ndx) ≠ (𝐸‘ndx) |
sralem.3 | ⊢ ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx) |
sralem.4 | ⊢ (·𝑖‘ndx) ≠ (𝐸‘ndx) |
Ref | Expression |
---|---|
sralem | ⊢ (𝜑 → (𝐸‘𝑊) = (𝐸‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sralem.1 | . . . . 5 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
2 | sralem.2 | . . . . . 6 ⊢ (Scalar‘ndx) ≠ (𝐸‘ndx) | |
3 | 2 | necomi 2996 | . . . . 5 ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx) |
4 | 1, 3 | setsnid 17129 | . . . 4 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉)) |
5 | sralem.3 | . . . . . 6 ⊢ ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx) | |
6 | 5 | necomi 2996 | . . . . 5 ⊢ (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx) |
7 | 1, 6 | setsnid 17129 | . . . 4 ⊢ (𝐸‘(𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉)) = (𝐸‘((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉)) |
8 | sralem.4 | . . . . . 6 ⊢ (·𝑖‘ndx) ≠ (𝐸‘ndx) | |
9 | 8 | necomi 2996 | . . . . 5 ⊢ (𝐸‘ndx) ≠ (·𝑖‘ndx) |
10 | 1, 9 | setsnid 17129 | . . . 4 ⊢ (𝐸‘((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉)) = (𝐸‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
11 | 4, 7, 10 | 3eqtri 2765 | . . 3 ⊢ (𝐸‘𝑊) = (𝐸‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
12 | srapart.a | . . . . . 6 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) | |
13 | 12 | adantl 483 | . . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
14 | srapart.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) | |
15 | sraval 20766 | . . . . . 6 ⊢ ((𝑊 ∈ V ∧ 𝑆 ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) | |
16 | 14, 15 | sylan2 594 | . . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
17 | 13, 16 | eqtrd 2773 | . . . 4 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
18 | 17 | fveq2d 6885 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝐴) = (𝐸‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉))) |
19 | 11, 18 | eqtr4id 2792 | . 2 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = (𝐸‘𝐴)) |
20 | 1 | str0 17109 | . . 3 ⊢ ∅ = (𝐸‘∅) |
21 | fvprc 6873 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = ∅) | |
22 | 21 | adantr 482 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = ∅) |
23 | fv2prc 6926 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = ∅) | |
24 | 12, 23 | sylan9eqr 2795 | . . . 4 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → 𝐴 = ∅) |
25 | 24 | fveq2d 6885 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝐴) = (𝐸‘∅)) |
26 | 20, 22, 25 | 3eqtr4a 2799 | . 2 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = (𝐸‘𝐴)) |
27 | 19, 26 | pm2.61ian 811 | 1 ⊢ (𝜑 → (𝐸‘𝑊) = (𝐸‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ⊆ wss 3946 ∅c0 4320 〈cop 4630 ‘cfv 6535 (class class class)co 7396 sSet csts 17083 Slot cslot 17101 ndxcnx 17113 Basecbs 17131 ↾s cress 17160 .rcmulr 17185 Scalarcsca 17187 ·𝑠 cvsca 17188 ·𝑖cip 17189 subringAlg csra 20758 |
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-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 |
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 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-ov 7399 df-oprab 7400 df-mpo 7401 df-sets 17084 df-slot 17102 df-sra 20762 |
This theorem is referenced by: srabase 20769 sraaddg 20771 sramulr 20773 sratset 20780 srads 20783 cchhllem 28111 |
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