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Theorem sralemOLD 20797

Description: Obsolete version of sralem 20796 as of 29-Oct-2024. Lemma for srabase 20798 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
srapart.a	⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
srapart.s	⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))
sralemOLD.1	⊢ 𝐸 = Slot 𝑁
sralemOLD.2	⊢ 𝑁 ∈ ℕ
sralemOLD.3	⊢ (𝑁 < 5 ∨ 8 < 𝑁)

**Assertion**
Ref	Expression
sralemOLD	⊢ (𝜑 → (𝐸‘𝑊) = (𝐸‘𝐴))

**Proof of Theorem sralemOLD**
Step	Hyp	Ref	Expression
1		sralemOLD.1	. . . . . 6 ⊢ 𝐸 = Slot 𝑁
2		sralemOLD.2	. . . . . 6 ⊢ 𝑁 ∈ ℕ
3	1, 2	ndxid 17132	. . . . 5 ⊢ 𝐸 = Slot (𝐸‘ndx)
4		sralemOLD.3	. . . . . . 7 ⊢ (𝑁 < 5 ∨ 8 < 𝑁)
5	2	nnrei 12223	. . . . . . . . . 10 ⊢ 𝑁 ∈ ℝ
6		5re 12301	. . . . . . . . . 10 ⊢ 5 ∈ ℝ
7	5, 6	ltnei 11340	. . . . . . . . 9 ⊢ (𝑁 < 5 → 5 ≠ 𝑁)
8	7	necomd 2996	. . . . . . . 8 ⊢ (𝑁 < 5 → 𝑁 ≠ 5)
9		5lt8 12408	. . . . . . . . . 10 ⊢ 5 < 8
10		8re 12310	. . . . . . . . . . 11 ⊢ 8 ∈ ℝ
11	6, 10, 5	lttri 11342	. . . . . . . . . 10 ⊢ ((5 < 8 ∧ 8 < 𝑁) → 5 < 𝑁)
12	9, 11	mpan 688	. . . . . . . . 9 ⊢ (8 < 𝑁 → 5 < 𝑁)
13	6, 5	ltnei 11340	. . . . . . . . 9 ⊢ (5 < 𝑁 → 𝑁 ≠ 5)
14	12, 13	syl 17	. . . . . . . 8 ⊢ (8 < 𝑁 → 𝑁 ≠ 5)
15	8, 14	jaoi 855	. . . . . . 7 ⊢ ((𝑁 < 5 ∨ 8 < 𝑁) → 𝑁 ≠ 5)
16	4, 15	ax-mp 5	. . . . . 6 ⊢ 𝑁 ≠ 5
17	1, 2	ndxarg 17131	. . . . . . 7 ⊢ (𝐸‘ndx) = 𝑁
18		scandx 17261	. . . . . . 7 ⊢ (Scalar‘ndx) = 5
19	17, 18	neeq12i 3007	. . . . . 6 ⊢ ((𝐸‘ndx) ≠ (Scalar‘ndx) ↔ 𝑁 ≠ 5)
20	16, 19	mpbir 230	. . . . 5 ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx)
21	3, 20	setsnid 17144	. . . 4 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾_s 𝑆)⟩))
22		5lt6 12395	. . . . . . . . . . 11 ⊢ 5 < 6
23		6re 12304	. . . . . . . . . . . 12 ⊢ 6 ∈ ℝ
24	5, 6, 23	lttri 11342	. . . . . . . . . . 11 ⊢ ((𝑁 < 5 ∧ 5 < 6) → 𝑁 < 6)
25	22, 24	mpan2 689	. . . . . . . . . 10 ⊢ (𝑁 < 5 → 𝑁 < 6)
26	5, 23	ltnei 11340	. . . . . . . . . 10 ⊢ (𝑁 < 6 → 6 ≠ 𝑁)
27	25, 26	syl 17	. . . . . . . . 9 ⊢ (𝑁 < 5 → 6 ≠ 𝑁)
28	27	necomd 2996	. . . . . . . 8 ⊢ (𝑁 < 5 → 𝑁 ≠ 6)
29		6lt8 12407	. . . . . . . . . 10 ⊢ 6 < 8
30	23, 10, 5	lttri 11342	. . . . . . . . . 10 ⊢ ((6 < 8 ∧ 8 < 𝑁) → 6 < 𝑁)
31	29, 30	mpan 688	. . . . . . . . 9 ⊢ (8 < 𝑁 → 6 < 𝑁)
32	23, 5	ltnei 11340	. . . . . . . . 9 ⊢ (6 < 𝑁 → 𝑁 ≠ 6)
33	31, 32	syl 17	. . . . . . . 8 ⊢ (8 < 𝑁 → 𝑁 ≠ 6)
34	28, 33	jaoi 855	. . . . . . 7 ⊢ ((𝑁 < 5 ∨ 8 < 𝑁) → 𝑁 ≠ 6)
35	4, 34	ax-mp 5	. . . . . 6 ⊢ 𝑁 ≠ 6
36		vscandx 17266	. . . . . . 7 ⊢ ( ·_𝑠 ‘ndx) = 6
37	17, 36	neeq12i 3007	. . . . . 6 ⊢ ((𝐸‘ndx) ≠ ( ·_𝑠 ‘ndx) ↔ 𝑁 ≠ 6)
38	35, 37	mpbir 230	. . . . 5 ⊢ (𝐸‘ndx) ≠ ( ·_𝑠 ‘ndx)
39	3, 38	setsnid 17144	. . . 4 ⊢ (𝐸‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾_s 𝑆)⟩)) = (𝐸‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾_s 𝑆)⟩) sSet ⟨( ·_𝑠 ‘ndx), (._r‘𝑊)⟩))
40	5, 6, 10	lttri 11342	. . . . . . . . . . 11 ⊢ ((𝑁 < 5 ∧ 5 < 8) → 𝑁 < 8)
41	9, 40	mpan2 689	. . . . . . . . . 10 ⊢ (𝑁 < 5 → 𝑁 < 8)
42	5, 10	ltnei 11340	. . . . . . . . . 10 ⊢ (𝑁 < 8 → 8 ≠ 𝑁)
43	41, 42	syl 17	. . . . . . . . 9 ⊢ (𝑁 < 5 → 8 ≠ 𝑁)
44	43	necomd 2996	. . . . . . . 8 ⊢ (𝑁 < 5 → 𝑁 ≠ 8)
45	10, 5	ltnei 11340	. . . . . . . 8 ⊢ (8 < 𝑁 → 𝑁 ≠ 8)
46	44, 45	jaoi 855	. . . . . . 7 ⊢ ((𝑁 < 5 ∨ 8 < 𝑁) → 𝑁 ≠ 8)
47	4, 46	ax-mp 5	. . . . . 6 ⊢ 𝑁 ≠ 8
48		ipndx 17277	. . . . . . 7 ⊢ (·_𝑖‘ndx) = 8
49	17, 48	neeq12i 3007	. . . . . 6 ⊢ ((𝐸‘ndx) ≠ (·_𝑖‘ndx) ↔ 𝑁 ≠ 8)
50	47, 49	mpbir 230	. . . . 5 ⊢ (𝐸‘ndx) ≠ (·_𝑖‘ndx)
51	3, 50	setsnid 17144	. . . 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)⟩))
52	21, 39, 51	3eqtri 2764	. . 3 ⊢ (𝐸‘𝑊) = (𝐸‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾_s 𝑆)⟩) sSet ⟨( ·_𝑠 ‘ndx), (._r‘𝑊)⟩) sSet ⟨(·_𝑖‘ndx), (._r‘𝑊)⟩))
53		srapart.a	. . . . . 6 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
54	53	adantl 482	. . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
55		srapart.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))
56		sraval 20795	. . . . . 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)⟩))
57	55, 56	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)⟩))
58	54, 57	eqtrd 2772	. . . 4 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾_s 𝑆)⟩) sSet ⟨( ·_𝑠 ‘ndx), (._r‘𝑊)⟩) sSet ⟨(·_𝑖‘ndx), (._r‘𝑊)⟩))
59	58	fveq2d 6895	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝐴) = (𝐸‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾_s 𝑆)⟩) sSet ⟨( ·_𝑠 ‘ndx), (._r‘𝑊)⟩) sSet ⟨(·_𝑖‘ndx), (._r‘𝑊)⟩)))
60	52, 59	eqtr4id 2791	. 2 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = (𝐸‘𝐴))
61	1	str0 17124	. . 3 ⊢ ∅ = (𝐸‘∅)
62		fvprc 6883	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = ∅)
63	62	adantr 481	. . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = ∅)
64		fv2prc 6936	. . . . 5 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = ∅)
65	53, 64	sylan9eqr 2794	. . . 4 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → 𝐴 = ∅)
66	65	fveq2d 6895	. . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝐴) = (𝐸‘∅))
67	61, 63, 66	3eqtr4a 2798	. 2 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = (𝐸‘𝐴))
68	60, 67	pm2.61ian 810	1 ⊢ (𝜑 → (𝐸‘𝑊) = (𝐸‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 Vcvv 3474 ⊆ wss 3948 ∅c0 4322 ⟨cop 4634 class class class wbr 5148 ‘cfv 6543 (class class class)co 7411 < clt 11250 ℕcn 12214 5c5 12272 6c6 12273 8c8 12275 sSet csts 17098 Slot cslot 17116 ndxcnx 17128 Basecbs 17146 ↾_s cress 17175 ._rcmulr 17200 Scalarcsca 17202 ·_𝑠 cvsca 17203 ·_𝑖cip 17204 subringAlg csra 20787

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-sets 17099 df-slot 17117 df-ndx 17129 df-sca 17215 df-vsca 17216 df-ip 17217 df-sra 20791

This theorem is referenced by: srabaseOLD 20799 sraaddgOLD 20801 sramulrOLD 20803 sratsetOLD 20810 sradsOLD 20813 cchhllemOLD 28183

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