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Theorem sralemOLD 20791

Description: Obsolete version of sralem 20790 as of 29-Oct-2024. Lemma for srabase 20792 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 17130	. . . . 5 ⊢ 𝐸 = Slot (𝐸‘ndx)
4		sralemOLD.3	. . . . . . 7 ⊢ (𝑁 < 5 ∨ 8 < 𝑁)
5	2	nnrei 12221	. . . . . . . . . 10 ⊢ 𝑁 ∈ ℝ
6		5re 12299	. . . . . . . . . 10 ⊢ 5 ∈ ℝ
7	5, 6	ltnei 11338	. . . . . . . . 9 ⊢ (𝑁 < 5 → 5 ≠ 𝑁)
8	7	necomd 2997	. . . . . . . 8 ⊢ (𝑁 < 5 → 𝑁 ≠ 5)
9		5lt8 12406	. . . . . . . . . 10 ⊢ 5 < 8
10		8re 12308	. . . . . . . . . . 11 ⊢ 8 ∈ ℝ
11	6, 10, 5	lttri 11340	. . . . . . . . . 10 ⊢ ((5 < 8 ∧ 8 < 𝑁) → 5 < 𝑁)
12	9, 11	mpan 689	. . . . . . . . 9 ⊢ (8 < 𝑁 → 5 < 𝑁)
13	6, 5	ltnei 11338	. . . . . . . . 9 ⊢ (5 < 𝑁 → 𝑁 ≠ 5)
14	12, 13	syl 17	. . . . . . . 8 ⊢ (8 < 𝑁 → 𝑁 ≠ 5)
15	8, 14	jaoi 856	. . . . . . 7 ⊢ ((𝑁 < 5 ∨ 8 < 𝑁) → 𝑁 ≠ 5)
16	4, 15	ax-mp 5	. . . . . 6 ⊢ 𝑁 ≠ 5
17	1, 2	ndxarg 17129	. . . . . . 7 ⊢ (𝐸‘ndx) = 𝑁
18		scandx 17259	. . . . . . 7 ⊢ (Scalar‘ndx) = 5
19	17, 18	neeq12i 3008	. . . . . 6 ⊢ ((𝐸‘ndx) ≠ (Scalar‘ndx) ↔ 𝑁 ≠ 5)
20	16, 19	mpbir 230	. . . . 5 ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx)
21	3, 20	setsnid 17142	. . . 4 ⊢ (𝐸‘𝑊) = (𝐸‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾_s 𝑆)⟩))
22		5lt6 12393	. . . . . . . . . . 11 ⊢ 5 < 6
23		6re 12302	. . . . . . . . . . . 12 ⊢ 6 ∈ ℝ
24	5, 6, 23	lttri 11340	. . . . . . . . . . 11 ⊢ ((𝑁 < 5 ∧ 5 < 6) → 𝑁 < 6)
25	22, 24	mpan2 690	. . . . . . . . . 10 ⊢ (𝑁 < 5 → 𝑁 < 6)
26	5, 23	ltnei 11338	. . . . . . . . . 10 ⊢ (𝑁 < 6 → 6 ≠ 𝑁)
27	25, 26	syl 17	. . . . . . . . 9 ⊢ (𝑁 < 5 → 6 ≠ 𝑁)
28	27	necomd 2997	. . . . . . . 8 ⊢ (𝑁 < 5 → 𝑁 ≠ 6)
29		6lt8 12405	. . . . . . . . . 10 ⊢ 6 < 8
30	23, 10, 5	lttri 11340	. . . . . . . . . 10 ⊢ ((6 < 8 ∧ 8 < 𝑁) → 6 < 𝑁)
31	29, 30	mpan 689	. . . . . . . . 9 ⊢ (8 < 𝑁 → 6 < 𝑁)
32	23, 5	ltnei 11338	. . . . . . . . 9 ⊢ (6 < 𝑁 → 𝑁 ≠ 6)
33	31, 32	syl 17	. . . . . . . 8 ⊢ (8 < 𝑁 → 𝑁 ≠ 6)
34	28, 33	jaoi 856	. . . . . . 7 ⊢ ((𝑁 < 5 ∨ 8 < 𝑁) → 𝑁 ≠ 6)
35	4, 34	ax-mp 5	. . . . . 6 ⊢ 𝑁 ≠ 6
36		vscandx 17264	. . . . . . 7 ⊢ ( ·_𝑠 ‘ndx) = 6
37	17, 36	neeq12i 3008	. . . . . 6 ⊢ ((𝐸‘ndx) ≠ ( ·_𝑠 ‘ndx) ↔ 𝑁 ≠ 6)
38	35, 37	mpbir 230	. . . . 5 ⊢ (𝐸‘ndx) ≠ ( ·_𝑠 ‘ndx)
39	3, 38	setsnid 17142	. . . 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 11340	. . . . . . . . . . 11 ⊢ ((𝑁 < 5 ∧ 5 < 8) → 𝑁 < 8)
41	9, 40	mpan2 690	. . . . . . . . . 10 ⊢ (𝑁 < 5 → 𝑁 < 8)
42	5, 10	ltnei 11338	. . . . . . . . . 10 ⊢ (𝑁 < 8 → 8 ≠ 𝑁)
43	41, 42	syl 17	. . . . . . . . 9 ⊢ (𝑁 < 5 → 8 ≠ 𝑁)
44	43	necomd 2997	. . . . . . . 8 ⊢ (𝑁 < 5 → 𝑁 ≠ 8)
45	10, 5	ltnei 11338	. . . . . . . 8 ⊢ (8 < 𝑁 → 𝑁 ≠ 8)
46	44, 45	jaoi 856	. . . . . . 7 ⊢ ((𝑁 < 5 ∨ 8 < 𝑁) → 𝑁 ≠ 8)
47	4, 46	ax-mp 5	. . . . . 6 ⊢ 𝑁 ≠ 8
48		ipndx 17275	. . . . . . 7 ⊢ (·_𝑖‘ndx) = 8
49	17, 48	neeq12i 3008	. . . . . 6 ⊢ ((𝐸‘ndx) ≠ (·_𝑖‘ndx) ↔ 𝑁 ≠ 8)
50	47, 49	mpbir 230	. . . . 5 ⊢ (𝐸‘ndx) ≠ (·_𝑖‘ndx)
51	3, 50	setsnid 17142	. . . 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 2765	. . 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 483	. . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
55		srapart.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))
56		sraval 20789	. . . . . 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 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)⟩))
58	54, 57	eqtrd 2773	. . . 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 6896	. . 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 2792	. 2 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = (𝐸‘𝐴))
61	1	str0 17122	. . 3 ⊢ ∅ = (𝐸‘∅)
62		fvprc 6884	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝐸‘𝑊) = ∅)
63	62	adantr 482	. . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = ∅)
64		fv2prc 6937	. . . . 5 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = ∅)
65	53, 64	sylan9eqr 2795	. . . 4 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → 𝐴 = ∅)
66	65	fveq2d 6896	. . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝐴) = (𝐸‘∅))
67	61, 63, 66	3eqtr4a 2799	. 2 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝐸‘𝑊) = (𝐸‘𝐴))
68	60, 67	pm2.61ian 811	1 ⊢ (𝜑 → (𝐸‘𝑊) = (𝐸‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ⊆ wss 3949 ∅c0 4323 ⟨cop 4635 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 < clt 11248 ℕcn 12212 5c5 12270 6c6 12271 8c8 12273 sSet csts 17096 Slot cslot 17114 ndxcnx 17126 Basecbs 17144 ↾_s cress 17173 ._rcmulr 17198 Scalarcsca 17200 ·_𝑠 cvsca 17201 ·_𝑖cip 17202 subringAlg csra 20781

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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-sets 17097 df-slot 17115 df-ndx 17127 df-sca 17213 df-vsca 17214 df-ip 17215 df-sra 20785

This theorem is referenced by: srabaseOLD 20793 sraaddgOLD 20795 sramulrOLD 20797 sratsetOLD 20804 sradsOLD 20807 cchhllemOLD 28145

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