Theorem stoweidlem23 46385

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 548	. . . . . . . . 9 ⊢ (𝜑 → (𝜑 ∧ 𝐺 ∈ 𝐴))
5		eleq1 2825	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐺 → (𝑓 ∈ 𝐴 ↔ 𝐺 ∈ 𝐴))
6	5	anbi2d 631	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐺 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐺 ∈ 𝐴)))
7		feq1 6648	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐺 → (𝑓:𝑇⟶ℝ ↔ 𝐺:𝑇⟶ℝ))
8	6, 7	imbi12d 344	. . . . . . . . . 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 3513	. . . . . . . . 9 ⊢ (𝐺 ∈ 𝐴 → ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐺:𝑇⟶ℝ))
11	3, 4, 10	sylc 65	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑇⟶ℝ)
12	11	ffvelcdmda 7038	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℝ)
13	12	recnd 11172	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℂ)
14		stoweidlem23.8	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ 𝑇)
15	11, 14	ffvelcdmd 7039	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝑍) ∈ ℝ)
16	15	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑍) ∈ ℝ)
17	16	recnd 11172	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑍) ∈ ℂ)
18	13, 17	negsubd 11510	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) + -(𝐺‘𝑍)) = ((𝐺‘𝑡) − (𝐺‘𝑍)))
19	2, 18	mpteq2da 5192	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) + -(𝐺‘𝑍))) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐺‘𝑍))))
20		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
21	15	renegcld 11576	. . . . . . . . 9 ⊢ (𝜑 → -(𝐺‘𝑍) ∈ ℝ)
22	21	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → -(𝐺‘𝑍) ∈ ℝ)
23		eqid 2737	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍)) = (𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))
24	23	fvmpt2 6961	. . . . . . . 8 ⊢ ((𝑡 ∈ 𝑇 ∧ -(𝐺‘𝑍) ∈ ℝ) → ((𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))‘𝑡) = -(𝐺‘𝑍))
25	20, 22, 24	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))‘𝑡) = -(𝐺‘𝑍))
26	25	oveq2d 7384	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐺‘𝑡) + ((𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))‘𝑡)) = ((𝐺‘𝑡) + -(𝐺‘𝑍)))
27	2, 26	mpteq2da 5192	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) + ((𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) + -(𝐺‘𝑍))))
28	21	ancli 548	. . . . . . 7 ⊢ (𝜑 → (𝜑 ∧ -(𝐺‘𝑍) ∈ ℝ))
29		eleq1 2825	. . . . . . . . . 10 ⊢ (𝑥 = -(𝐺‘𝑍) → (𝑥 ∈ ℝ ↔ -(𝐺‘𝑍) ∈ ℝ))
30	29	anbi2d 631	. . . . . . . . 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 6852	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡(𝐺‘𝑍)
34	33	nfneg 11388	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡-(𝐺‘𝑍)
35	34	nfeq2 2917	. . . . . . . . . . 11 ⊢ Ⅎ𝑡 𝑥 = -(𝐺‘𝑍)
36		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑥 = -(𝐺‘𝑍) ∧ 𝑡 ∈ 𝑇) → 𝑥 = -(𝐺‘𝑍))
37	35, 36	mpteq2da 5192	. . . . . . . . . 10 ⊢ (𝑥 = -(𝐺‘𝑍) → (𝑡 ∈ 𝑇 ↦ 𝑥) = (𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍)))
38	37	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑥 = -(𝐺‘𝑍) → ((𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍)) ∈ 𝐴))
39	30, 38	imbi12d 344	. . . . . . . 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 3513	. . . . . . 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 5199	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))
45	43, 31, 44	stoweidlem8 46370	. . . . . 6 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ (𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) + ((𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))‘𝑡))) ∈ 𝐴)
46	3, 42, 45	mpd3an23 1466	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) + ((𝑡 ∈ 𝑇 ↦ -(𝐺‘𝑍))‘𝑡))) ∈ 𝐴)
47	27, 46	eqeltrrd 2838	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) + -(𝐺‘𝑍))) ∈ 𝐴)
48	19, 47	eqeltrrd 2838	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐺‘𝑍))) ∈ 𝐴)
49	1, 48	eqeltrid 2841	. 2 ⊢ (𝜑 → 𝐻 ∈ 𝐴)
50		stoweidlem23.7	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝑇)
51	11, 50	ffvelcdmd 7039	. . . . 5 ⊢ (𝜑 → (𝐺‘𝑆) ∈ ℝ)
52	51	recnd 11172	. . . 4 ⊢ (𝜑 → (𝐺‘𝑆) ∈ ℂ)
53	15	recnd 11172	. . . 4 ⊢ (𝜑 → (𝐺‘𝑍) ∈ ℂ)
54		stoweidlem23.10	. . . 4 ⊢ (𝜑 → (𝐺‘𝑆) ≠ (𝐺‘𝑍))
55	52, 53, 54	subne0d 11513	. . 3 ⊢ (𝜑 → ((𝐺‘𝑆) − (𝐺‘𝑍)) ≠ 0)
56	51, 15	resubcld 11577	. . . 4 ⊢ (𝜑 → ((𝐺‘𝑆) − (𝐺‘𝑍)) ∈ ℝ)
57		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑡𝑆
58	31, 57	nffv 6852	. . . . . 6 ⊢ Ⅎ𝑡(𝐺‘𝑆)
59		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑡 −
60	58, 59, 33	nfov 7398	. . . . 5 ⊢ Ⅎ𝑡((𝐺‘𝑆) − (𝐺‘𝑍))
61		fveq2 6842	. . . . . 6 ⊢ (𝑡 = 𝑆 → (𝐺‘𝑡) = (𝐺‘𝑆))
62	61	oveq1d 7383	. . . . 5 ⊢ (𝑡 = 𝑆 → ((𝐺‘𝑡) − (𝐺‘𝑍)) = ((𝐺‘𝑆) − (𝐺‘𝑍)))
63	57, 60, 62, 1	fvmptf 6971	. . . 4 ⊢ ((𝑆 ∈ 𝑇 ∧ ((𝐺‘𝑆) − (𝐺‘𝑍)) ∈ ℝ) → (𝐻‘𝑆) = ((𝐺‘𝑆) − (𝐺‘𝑍)))
64	50, 56, 63	syl2anc 585	. . 3 ⊢ (𝜑 → (𝐻‘𝑆) = ((𝐺‘𝑆) − (𝐺‘𝑍)))
65	15, 15	resubcld 11577	. . . . 5 ⊢ (𝜑 → ((𝐺‘𝑍) − (𝐺‘𝑍)) ∈ ℝ)
66	33, 59, 33	nfov 7398	. . . . . 6 ⊢ Ⅎ𝑡((𝐺‘𝑍) − (𝐺‘𝑍))
67		fveq2 6842	. . . . . . 7 ⊢ (𝑡 = 𝑍 → (𝐺‘𝑡) = (𝐺‘𝑍))
68	67	oveq1d 7383	. . . . . 6 ⊢ (𝑡 = 𝑍 → ((𝐺‘𝑡) − (𝐺‘𝑍)) = ((𝐺‘𝑍) − (𝐺‘𝑍)))
69	32, 66, 68, 1	fvmptf 6971	. . . . 5 ⊢ ((𝑍 ∈ 𝑇 ∧ ((𝐺‘𝑍) − (𝐺‘𝑍)) ∈ ℝ) → (𝐻‘𝑍) = ((𝐺‘𝑍) − (𝐺‘𝑍)))
70	14, 65, 69	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝐻‘𝑍) = ((𝐺‘𝑍) − (𝐺‘𝑍)))
71	53	subidd 11492	. . . 4 ⊢ (𝜑 → ((𝐺‘𝑍) − (𝐺‘𝑍)) = 0)
72	70, 71	eqtrd 2772	. . 3 ⊢ (𝜑 → (𝐻‘𝑍) = 0)
73	55, 64, 72	3netr4d 3010	. 2 ⊢ (𝜑 → (𝐻‘𝑆) ≠ (𝐻‘𝑍))
74	49, 73, 72	3jca 1129	1 ⊢ (𝜑 → (𝐻 ∈ 𝐴 ∧ (𝐻‘𝑆) ≠ (𝐻‘𝑍) ∧ (𝐻‘𝑍) = 0))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114  Ⅎwnfc 2884   ≠ wne 2933   ↦ cmpt 5181  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  ℝcr 11037  0cc0 11038   + caddc 11041   − cmin 11376  -cneg 11377

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-po 5540  df-so 5541  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-ltxr 11183  df-sub 11378  df-neg 11379

This theorem is referenced by:  stoweidlem43  46405