Theorem stoweidlem23 40240

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 574	. . . . . . . . 9 ⊢ (𝜑 → (𝜑 ∧ 𝐺 ∈ 𝐴))
5		eleq1 2689	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐺 → (𝑓 ∈ 𝐴 ↔ 𝐺 ∈ 𝐴))
6	5	anbi2d 740	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐺 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐺 ∈ 𝐴)))
7		feq1 6026	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐺 → (𝑓:𝑇⟶ℝ ↔ 𝐺:𝑇⟶ℝ))
8	6, 7	imbi12d 334	. . . . . . . . . 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 3266	. . . . . . . . 9 ⊢ (𝐺 ∈ 𝐴 → ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐺:𝑇⟶ℝ))
11	3, 4, 10	sylc 65	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑇⟶ℝ)
12	11	ffvelrnda 6359	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℝ)
13	12	recnd 10068	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℂ)
14		stoweidlem23.8	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ 𝑇)
15	11, 14	ffvelrnd 6360	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝑍) ∈ ℝ)
16	15	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑍) ∈ ℝ)
17	16	recnd 10068	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑍) ∈ ℂ)
18	13, 17	negsubd 10398	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) + -(𝐺‘𝑍)) = ((𝐺‘𝑡) − (𝐺‘𝑍)))
19	2, 18	mpteq2da 4743	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) + -(𝐺‘𝑍))) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐺‘𝑍))))
20		simpr 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
21	15	renegcld 10457	. . . . . . . . 9 ⊢ (𝜑 → -(𝐺‘𝑍) ∈ ℝ)
22	21	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → -(𝐺‘𝑍) ∈ ℝ)
23		eqid 2622	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍)) = (𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))
24	23	fvmpt2 6291	. . . . . . . 8 ⊢ ((𝑡 ∈ 𝑇 ∧ -(𝐺‘𝑍) ∈ ℝ) → ((𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))‘𝑡) = -(𝐺‘𝑍))
25	20, 22, 24	syl2anc 693	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))‘𝑡) = -(𝐺‘𝑍))
26	25	oveq2d 6666	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) + ((𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))‘𝑡)) = ((𝐺‘𝑡) + -(𝐺‘𝑍)))
27	2, 26	mpteq2da 4743	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) + ((𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) + -(𝐺‘𝑍))))
28	21	ancli 574	. . . . . . 7 ⊢ (𝜑 → (𝜑 ∧ -(𝐺‘𝑍) ∈ ℝ))
29		eleq1 2689	. . . . . . . . . 10 ⊢ (𝑥 = -(𝐺‘𝑍) → (𝑥 ∈ ℝ ↔ -(𝐺‘𝑍) ∈ ℝ))
30	29	anbi2d 740	. . . . . . . . 9 ⊢ (𝑥 = -(𝐺‘𝑍) → ((𝜑 ∧ 𝑥 ∈ ℝ) ↔ (𝜑 ∧ -(𝐺‘𝑍) ∈ ℝ)))
31		stoweidlem23.2	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡𝐺
32		nfcv 2764	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡𝑍
33	31, 32	nffv 6198	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡(𝐺‘𝑍)
34	33	nfneg 10277	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡-(𝐺‘𝑍)
35	34	nfeq2 2780	. . . . . . . . . . 11 ⊢ Ⅎ𝑡 𝑥 = -(𝐺‘𝑍)
36		simpl 473	. . . . . . . . . . 11 ⊢ ((𝑥 = -(𝐺‘𝑍) ∧ 𝑡 ∈ 𝑇) → 𝑥 = -(𝐺‘𝑍))
37	35, 36	mpteq2da 4743	. . . . . . . . . 10 ⊢ (𝑥 = -(𝐺‘𝑍) → (𝑡 ∈ 𝑇 ↦ 𝑥) = (𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍)))
38	37	eleq1d 2686	. . . . . . . . 9 ⊢ (𝑥 = -(𝐺‘𝑍) → ((𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍)) ∈ 𝐴))
39	30, 38	imbi12d 334	. . . . . . . 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 3266	. . . . . . 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 4747	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))
45	43, 31, 44	stoweidlem8 40225	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ (𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) + ((𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))‘𝑡))) ∈ 𝐴)
46	3, 42, 45	mpd3an23 1426	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) + ((𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))‘𝑡))) ∈ 𝐴)
47	27, 46	eqeltrrd 2702	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) + -(𝐺‘𝑍))) ∈ 𝐴)
48	19, 47	eqeltrrd 2702	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐺‘𝑍))) ∈ 𝐴)
49	1, 48	syl5eqel 2705	. 2 ⊢ (𝜑 → 𝐻 ∈ 𝐴)
50		stoweidlem23.7	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝑇)
51	11, 50	ffvelrnd 6360	. . . . 5 ⊢ (𝜑 → (𝐺‘𝑆) ∈ ℝ)
52	51	recnd 10068	. . . 4 ⊢ (𝜑 → (𝐺‘𝑆) ∈ ℂ)
53	15	recnd 10068	. . . 4 ⊢ (𝜑 → (𝐺‘𝑍) ∈ ℂ)
54		stoweidlem23.10	. . . 4 ⊢ (𝜑 → (𝐺‘𝑆) ≠ (𝐺‘𝑍))
55	52, 53, 54	subne0d 10401	. . 3 ⊢ (𝜑 → ((𝐺‘𝑆) − (𝐺‘𝑍)) ≠ 0)
56	51, 15	resubcld 10458	. . . 4 ⊢ (𝜑 → ((𝐺‘𝑆) − (𝐺‘𝑍)) ∈ ℝ)
57		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑡𝑆
58	31, 57	nffv 6198	. . . . . 6 ⊢ Ⅎ𝑡(𝐺‘𝑆)
59		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑡 −
60	58, 59, 33	nfov 6676	. . . . 5 ⊢ Ⅎ𝑡((𝐺‘𝑆) − (𝐺‘𝑍))
61		fveq2 6191	. . . . . 6 ⊢ (𝑡 = 𝑆 → (𝐺‘𝑡) = (𝐺‘𝑆))
62	61	oveq1d 6665	. . . . 5 ⊢ (𝑡 = 𝑆 → ((𝐺‘𝑡) − (𝐺‘𝑍)) = ((𝐺‘𝑆) − (𝐺‘𝑍)))
63	57, 60, 62, 1	fvmptf 6301	. . . 4 ⊢ ((𝑆 ∈ 𝑇 ∧ ((𝐺‘𝑆) − (𝐺‘𝑍)) ∈ ℝ) → (𝐻‘𝑆) = ((𝐺‘𝑆) − (𝐺‘𝑍)))
64	50, 56, 63	syl2anc 693	. . 3 ⊢ (𝜑 → (𝐻‘𝑆) = ((𝐺‘𝑆) − (𝐺‘𝑍)))
65	15, 15	resubcld 10458	. . . . 5 ⊢ (𝜑 → ((𝐺‘𝑍) − (𝐺‘𝑍)) ∈ ℝ)
66	33, 59, 33	nfov 6676	. . . . . 6 ⊢ Ⅎ𝑡((𝐺‘𝑍) − (𝐺‘𝑍))
67		fveq2 6191	. . . . . . 7 ⊢ (𝑡 = 𝑍 → (𝐺‘𝑡) = (𝐺‘𝑍))
68	67	oveq1d 6665	. . . . . 6 ⊢ (𝑡 = 𝑍 → ((𝐺‘𝑡) − (𝐺‘𝑍)) = ((𝐺‘𝑍) − (𝐺‘𝑍)))
69	32, 66, 68, 1	fvmptf 6301	. . . . 5 ⊢ ((𝑍 ∈ 𝑇 ∧ ((𝐺‘𝑍) − (𝐺‘𝑍)) ∈ ℝ) → (𝐻‘𝑍) = ((𝐺‘𝑍) − (𝐺‘𝑍)))
70	14, 65, 69	syl2anc 693	. . . 4 ⊢ (𝜑 → (𝐻‘𝑍) = ((𝐺‘𝑍) − (𝐺‘𝑍)))
71	53	subidd 10380	. . . 4 ⊢ (𝜑 → ((𝐺‘𝑍) − (𝐺‘𝑍)) = 0)
72	70, 71	eqtrd 2656	. . 3 ⊢ (𝜑 → (𝐻‘𝑍) = 0)
73	55, 64, 72	3netr4d 2871	. 2 ⊢ (𝜑 → (𝐻‘𝑆) ≠ (𝐻‘𝑍))
74	49, 73, 72	3jca 1242	1 ⊢ (𝜑 → (𝐻 ∈ 𝐴 ∧ (𝐻‘𝑆) ≠ (𝐻‘𝑍) ∧ (𝐻‘𝑍) = 0))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  Ⅎwnfc 2751   ≠ wne 2794   ↦ cmpt 4729  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ℝcr 9935  0cc0 9936   + caddc 9939   − cmin 10266  -cneg 10267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-neg 10269

This theorem is referenced by:  stoweidlem43  40260