Theorem stoweidlem23 45440

Description: This lemma is used to prove the existence of a function p_t as in the beginning of Lemma 1 [BrosowskiDeutsh] p. 90: for all t in T - U, there exists a function p in the subalgebra, such that p_t ( t₀ ) = 0 , p_t ( t ) > 0, and 0 <= p_t <= 1. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem23.1	⊢ Ⅎ𝑡𝜑
stoweidlem23.2	⊢ Ⅎ𝑡𝐺
stoweidlem23.3	⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐺‘𝑍)))
stoweidlem23.4	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem23.5	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem23.6	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
stoweidlem23.7	⊢ (𝜑 → 𝑆 ∈ 𝑇)
stoweidlem23.8	⊢ (𝜑 → 𝑍 ∈ 𝑇)
stoweidlem23.9	⊢ (𝜑 → 𝐺 ∈ 𝐴)
stoweidlem23.10	⊢ (𝜑 → (𝐺‘𝑆) ≠ (𝐺‘𝑍))

Assertion

Ref	Expression
stoweidlem23	⊢ (𝜑 → (𝐻 ∈ 𝐴 ∧ (𝐻‘𝑆) ≠ (𝐻‘𝑍) ∧ (𝐻‘𝑍) = 0))

Distinct variable groups:   𝑓,𝑔,𝑡,𝑇   𝐴,𝑓,𝑔   𝑓,𝐺,𝑔   𝜑,𝑓,𝑔   𝑔,𝑍,𝑡   𝑥,𝑡,𝑇   𝑡,𝑆   𝑥,𝐴   𝑥,𝐺   𝑥,𝑍   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑡)   𝐴(𝑡)   𝑆(𝑥,𝑓,𝑔)   𝐺(𝑡)   𝐻(𝑥,𝑡,𝑓,𝑔)   𝑍(𝑓)

Proof of Theorem stoweidlem23

Step	Hyp	Ref	Expression
1		stoweidlem23.3	. . 3 ⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐺‘𝑍)))
2		stoweidlem23.1	. . . . 5 ⊢ Ⅎ𝑡𝜑
3		stoweidlem23.9	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝐴)
4	3	ancli 547	. . . . . . . . 9 ⊢ (𝜑 → (𝜑 ∧ 𝐺 ∈ 𝐴))
5		eleq1 2817	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐺 → (𝑓 ∈ 𝐴 ↔ 𝐺 ∈ 𝐴))
6	5	anbi2d 628	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐺 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐺 ∈ 𝐴)))
7		feq1 6708	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐺 → (𝑓:𝑇⟶ℝ ↔ 𝐺:𝑇⟶ℝ))
8	6, 7	imbi12d 343	. . . . . . . . . 10 ⊢ (𝑓 = 𝐺 → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐺:𝑇⟶ℝ)))
9		stoweidlem23.4	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
10	8, 9	vtoclg 3542	. . . . . . . . 9 ⊢ (𝐺 ∈ 𝐴 → ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐺:𝑇⟶ℝ))
11	3, 4, 10	sylc 65	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑇⟶ℝ)
12	11	ffvelcdmda 7099	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℝ)
13	12	recnd 11280	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℂ)
14		stoweidlem23.8	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ 𝑇)
15	11, 14	ffvelcdmd 7100	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝑍) ∈ ℝ)
16	15	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑍) ∈ ℝ)
17	16	recnd 11280	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑍) ∈ ℂ)
18	13, 17	negsubd 11615	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) + -(𝐺‘𝑍)) = ((𝐺‘𝑡) − (𝐺‘𝑍)))
19	2, 18	mpteq2da 5250	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) + -(𝐺‘𝑍))) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐺‘𝑍))))
20		simpr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
21	15	renegcld 11679	. . . . . . . . 9 ⊢ (𝜑 → -(𝐺‘𝑍) ∈ ℝ)
22	21	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → -(𝐺‘𝑍) ∈ ℝ)
23		eqid 2728	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍)) = (𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))
24	23	fvmpt2 7021	. . . . . . . 8 ⊢ ((𝑡 ∈ 𝑇 ∧ -(𝐺‘𝑍) ∈ ℝ) → ((𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))‘𝑡) = -(𝐺‘𝑍))
25	20, 22, 24	syl2anc 582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))‘𝑡) = -(𝐺‘𝑍))
26	25	oveq2d 7442	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) + ((𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))‘𝑡)) = ((𝐺‘𝑡) + -(𝐺‘𝑍)))
27	2, 26	mpteq2da 5250	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) + ((𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) + -(𝐺‘𝑍))))
28	21	ancli 547	. . . . . . 7 ⊢ (𝜑 → (𝜑 ∧ -(𝐺‘𝑍) ∈ ℝ))
29		eleq1 2817	. . . . . . . . . 10 ⊢ (𝑥 = -(𝐺‘𝑍) → (𝑥 ∈ ℝ ↔ -(𝐺‘𝑍) ∈ ℝ))
30	29	anbi2d 628	. . . . . . . . 9 ⊢ (𝑥 = -(𝐺‘𝑍) → ((𝜑 ∧ 𝑥 ∈ ℝ) ↔ (𝜑 ∧ -(𝐺‘𝑍) ∈ ℝ)))
31		stoweidlem23.2	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡𝐺
32		nfcv 2899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡𝑍
33	31, 32	nffv 6912	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡(𝐺‘𝑍)
34	33	nfneg 11494	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡-(𝐺‘𝑍)
35	34	nfeq2 2917	. . . . . . . . . . 11 ⊢ Ⅎ𝑡 𝑥 = -(𝐺‘𝑍)
36		simpl 481	. . . . . . . . . . 11 ⊢ ((𝑥 = -(𝐺‘𝑍) ∧ 𝑡 ∈ 𝑇) → 𝑥 = -(𝐺‘𝑍))
37	35, 36	mpteq2da 5250	. . . . . . . . . 10 ⊢ (𝑥 = -(𝐺‘𝑍) → (𝑡 ∈ 𝑇 ↦ 𝑥) = (𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍)))
38	37	eleq1d 2814	. . . . . . . . 9 ⊢ (𝑥 = -(𝐺‘𝑍) → ((𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍)) ∈ 𝐴))
39	30, 38	imbi12d 343	. . . . . . . 8 ⊢ (𝑥 = -(𝐺‘𝑍) → (((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) ↔ ((𝜑 ∧ -(𝐺‘𝑍) ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍)) ∈ 𝐴)))
40		stoweidlem23.6	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
41	39, 40	vtoclg 3542	. . . . . . 7 ⊢ (-(𝐺‘𝑍) ∈ ℝ → ((𝜑 ∧ -(𝐺‘𝑍) ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍)) ∈ 𝐴))
42	21, 28, 41	sylc 65	. . . . . 6 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍)) ∈ 𝐴)
43		stoweidlem23.5	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
44		nfmpt1 5260	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))
45	43, 31, 44	stoweidlem8 45425	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ (𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) + ((𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))‘𝑡))) ∈ 𝐴)
46	3, 42, 45	mpd3an23 1459	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) + ((𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))‘𝑡))) ∈ 𝐴)
47	27, 46	eqeltrrd 2830	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) + -(𝐺‘𝑍))) ∈ 𝐴)
48	19, 47	eqeltrrd 2830	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐺‘𝑍))) ∈ 𝐴)
49	1, 48	eqeltrid 2833	. 2 ⊢ (𝜑 → 𝐻 ∈ 𝐴)
50		stoweidlem23.7	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝑇)
51	11, 50	ffvelcdmd 7100	. . . . 5 ⊢ (𝜑 → (𝐺‘𝑆) ∈ ℝ)
52	51	recnd 11280	. . . 4 ⊢ (𝜑 → (𝐺‘𝑆) ∈ ℂ)
53	15	recnd 11280	. . . 4 ⊢ (𝜑 → (𝐺‘𝑍) ∈ ℂ)
54		stoweidlem23.10	. . . 4 ⊢ (𝜑 → (𝐺‘𝑆) ≠ (𝐺‘𝑍))
55	52, 53, 54	subne0d 11618	. . 3 ⊢ (𝜑 → ((𝐺‘𝑆) − (𝐺‘𝑍)) ≠ 0)
56	51, 15	resubcld 11680	. . . 4 ⊢ (𝜑 → ((𝐺‘𝑆) − (𝐺‘𝑍)) ∈ ℝ)
57		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑡𝑆
58	31, 57	nffv 6912	. . . . . 6 ⊢ Ⅎ𝑡(𝐺‘𝑆)
59		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑡 −
60	58, 59, 33	nfov 7456	. . . . 5 ⊢ Ⅎ𝑡((𝐺‘𝑆) − (𝐺‘𝑍))
61		fveq2 6902	. . . . . 6 ⊢ (𝑡 = 𝑆 → (𝐺‘𝑡) = (𝐺‘𝑆))
62	61	oveq1d 7441	. . . . 5 ⊢ (𝑡 = 𝑆 → ((𝐺‘𝑡) − (𝐺‘𝑍)) = ((𝐺‘𝑆) − (𝐺‘𝑍)))
63	57, 60, 62, 1	fvmptf 7031	. . . 4 ⊢ ((𝑆 ∈ 𝑇 ∧ ((𝐺‘𝑆) − (𝐺‘𝑍)) ∈ ℝ) → (𝐻‘𝑆) = ((𝐺‘𝑆) − (𝐺‘𝑍)))
64	50, 56, 63	syl2anc 582	. . 3 ⊢ (𝜑 → (𝐻‘𝑆) = ((𝐺‘𝑆) − (𝐺‘𝑍)))
65	15, 15	resubcld 11680	. . . . 5 ⊢ (𝜑 → ((𝐺‘𝑍) − (𝐺‘𝑍)) ∈ ℝ)
66	33, 59, 33	nfov 7456	. . . . . 6 ⊢ Ⅎ𝑡((𝐺‘𝑍) − (𝐺‘𝑍))
67		fveq2 6902	. . . . . . 7 ⊢ (𝑡 = 𝑍 → (𝐺‘𝑡) = (𝐺‘𝑍))
68	67	oveq1d 7441	. . . . . 6 ⊢ (𝑡 = 𝑍 → ((𝐺‘𝑡) − (𝐺‘𝑍)) = ((𝐺‘𝑍) − (𝐺‘𝑍)))
69	32, 66, 68, 1	fvmptf 7031	. . . . 5 ⊢ ((𝑍 ∈ 𝑇 ∧ ((𝐺‘𝑍) − (𝐺‘𝑍)) ∈ ℝ) → (𝐻‘𝑍) = ((𝐺‘𝑍) − (𝐺‘𝑍)))
70	14, 65, 69	syl2anc 582	. . . 4 ⊢ (𝜑 → (𝐻‘𝑍) = ((𝐺‘𝑍) − (𝐺‘𝑍)))
71	53	subidd 11597	. . . 4 ⊢ (𝜑 → ((𝐺‘𝑍) − (𝐺‘𝑍)) = 0)
72	70, 71	eqtrd 2768	. . 3 ⊢ (𝜑 → (𝐻‘𝑍) = 0)
73	55, 64, 72	3netr4d 3015	. 2 ⊢ (𝜑 → (𝐻‘𝑆) ≠ (𝐻‘𝑍))
74	49, 73, 72	3jca 1125	1 ⊢ (𝜑 → (𝐻 ∈ 𝐴 ∧ (𝐻‘𝑆) ≠ (𝐻‘𝑍) ∧ (𝐻‘𝑍) = 0))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533  Ⅎwnf 1777   ∈ wcel 2098  Ⅎwnfc 2879   ≠ wne 2937   ↦ cmpt 5235  ⟶wf 6549  ‘cfv 6553  (class class class)co 7426  ℝcr 11145  0cc0 11146   + caddc 11149   − cmin 11482  -cneg 11483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-po 5594  df-so 5595  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-pnf 11288  df-mnf 11289  df-ltxr 11291  df-sub 11484  df-neg 11485

This theorem is referenced by:  stoweidlem43  45460