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| Description: Lemma for srabase 21177 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 2995 | . . . . 5 ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx) | 
| 4 | 1, 3 | setsnid 17245 | . . . 4 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉)) | 
| 5 | sralem.3 | . . . . . 6 ⊢ ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx) | |
| 6 | 5 | necomi 2995 | . . . . 5 ⊢ (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx) | 
| 7 | 1, 6 | setsnid 17245 | . . . 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 2995 | . . . . 5 ⊢ (𝐸‘ndx) ≠ (·𝑖‘ndx) | 
| 10 | 1, 9 | setsnid 17245 | . . . 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 2769 | . . 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 21174 | . . . . . 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 2777 | . . . 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 6910 | . . 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 2796 | . 2 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = (𝐸‘𝐴)) | 
| 20 | 1 | str0 17226 | . . 3 ⊢ ∅ = (𝐸‘∅) | 
| 21 | fvprc 6898 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = ∅) | |
| 22 | 21 | adantr 480 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = ∅) | 
| 23 | fv2prc 6951 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = ∅) | |
| 24 | 12, 23 | sylan9eqr 2799 | . . . 4 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → 𝐴 = ∅) | 
| 25 | 24 | fveq2d 6910 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝐴) = (𝐸‘∅)) | 
| 26 | 20, 22, 25 | 3eqtr4a 2803 | . 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 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 〈cop 4632 ‘cfv 6561 (class class class)co 7431 sSet csts 17200 Slot cslot 17218 ndxcnx 17230 Basecbs 17247 ↾s cress 17274 .rcmulr 17298 Scalarcsca 17300 ·𝑠 cvsca 17301 ·𝑖cip 17302 subringAlg csra 21170 | 
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-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-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-id 5578 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-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-sets 17201 df-slot 17219 df-sra 21172 | 
| This theorem is referenced by: srabase 21177 sraaddg 21179 sramulr 21181 sratset 21188 srads 21191 cchhllem 28901 | 
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