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Mirrors > Home > MPE Home > Th. List > sralem | Structured version Visualization version GIF version |
Description: Lemma for srabase 21195 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 2993 | . . . . 5 ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx) |
4 | 1, 3 | setsnid 17243 | . . . 4 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉)) |
5 | sralem.3 | . . . . . 6 ⊢ ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx) | |
6 | 5 | necomi 2993 | . . . . 5 ⊢ (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx) |
7 | 1, 6 | setsnid 17243 | . . . 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 2993 | . . . . 5 ⊢ (𝐸‘ndx) ≠ (·𝑖‘ndx) |
10 | 1, 9 | setsnid 17243 | . . . 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 2767 | . . 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 481 | . . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
14 | srapart.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) | |
15 | sraval 21192 | . . . . . 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 593 | . . . . 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 2775 | . . . 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 6911 | . . 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 2794 | . 2 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = (𝐸‘𝐴)) |
20 | 1 | str0 17223 | . . 3 ⊢ ∅ = (𝐸‘∅) |
21 | fvprc 6899 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = ∅) | |
22 | 21 | adantr 480 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = ∅) |
23 | fv2prc 6952 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = ∅) | |
24 | 12, 23 | sylan9eqr 2797 | . . . 4 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → 𝐴 = ∅) |
25 | 24 | fveq2d 6911 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝐴) = (𝐸‘∅)) |
26 | 20, 22, 25 | 3eqtr4a 2801 | . 2 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = (𝐸‘𝐴)) |
27 | 19, 26 | pm2.61ian 812 | 1 ⊢ (𝜑 → (𝐸‘𝑊) = (𝐸‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 〈cop 4637 ‘cfv 6563 (class class class)co 7431 sSet csts 17197 Slot cslot 17215 ndxcnx 17227 Basecbs 17245 ↾s cress 17274 .rcmulr 17299 Scalarcsca 17301 ·𝑠 cvsca 17302 ·𝑖cip 17303 subringAlg csra 21188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-sets 17198 df-slot 17216 df-sra 21190 |
This theorem is referenced by: srabase 21195 sraaddg 21197 sramulr 21199 sratset 21206 srads 21209 cchhllem 28916 |
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