erdszelem11 - Mathbox for Mario Carneiro

	Mathbox for Mario Carneiro		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > erdszelem11		Structured version Visualization version GIF version

Theorem erdszelem11 35344

Description: Lemma for erdsze 35345. (Contributed by Mario Carneiro, 22-Jan-2015.)

**Hypotheses**
Ref	Expression
erdsze.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
erdsze.f	⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)
erdszelem.i	⊢ 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
erdszelem.j	⊢ 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ^◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
erdszelem.t	⊢ 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩)
erdszelem.r	⊢ (𝜑 → 𝑅 ∈ ℕ)
erdszelem.s	⊢ (𝜑 → 𝑆 ∈ ℕ)
erdszelem.m	⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁)

**Assertion**
Ref	Expression
erdszelem11	⊢ (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)))))

Distinct variable groups: 𝑥,𝑦 𝑛,𝑠,𝑥,𝑦,𝐹 𝑛,𝐼,𝑠,𝑥,𝑦 𝑛,𝐽,𝑠,𝑥,𝑦 𝑅,𝑠,𝑥,𝑦 𝑛,𝑁,𝑠,𝑥,𝑦 𝜑,𝑛,𝑠,𝑥,𝑦 𝑆,𝑠,𝑥,𝑦 𝑇,𝑠 Allowed substitution hints: 𝑅(𝑛) 𝑆(𝑛) 𝑇(𝑥,𝑦,𝑛)

Proof of Theorem erdszelem11 Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		erdsze.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		erdsze.f	. . . 4 ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)
3		erdszelem.i	. . . 4 ⊢ 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
4		erdszelem.j	. . . 4 ⊢ 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ^◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
5		erdszelem.t	. . . 4 ⊢ 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩)
6		erdszelem.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℕ)
7		erdszelem.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℕ)
8		erdszelem.m	. . . 4 ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁)
9	1, 2, 3, 4, 5, 6, 7, 8	erdszelem10 35343	. . 3 ⊢ (𝜑 → ∃𝑚 ∈ (1...𝑁)(¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1))))
10	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)))) → 𝑁 ∈ ℕ)
11	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)))) → 𝐹:(1...𝑁)–1-1→ℝ)
12		ltso 11211	. . . . . . 7 ⊢ < Or ℝ
13		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)))) → 𝑚 ∈ (1...𝑁))
14	6	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)))) → 𝑅 ∈ ℕ)
15		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)))) → ¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)))
16	10, 11, 3, 12, 13, 14, 15	erdszelem7 35340	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)))) → ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))))
17	16	expr 456	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → (¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)) → ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠)))))
18	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)))) → 𝑁 ∈ ℕ)
19	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)))) → 𝐹:(1...𝑁)–1-1→ℝ)
20		gtso 11212	. . . . . . 7 ⊢ ^◡ < Or ℝ
21		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)))) → 𝑚 ∈ (1...𝑁))
22	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)))) → 𝑆 ∈ ℕ)
23		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)))) → ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)))
24	18, 19, 4, 20, 21, 22, 23	erdszelem7 35340	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)))) → ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠))))
25	24	expr 456	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → (¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)) → ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)))))
26	17, 25	orim12d 966	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → ((¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1))) → (∃𝑠 ∈ 𝒫 (1...𝑁)(𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠))))))
27	26	rexlimdva 3135	. . 3 ⊢ (𝜑 → (∃𝑚 ∈ (1...𝑁)(¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1))) → (∃𝑠 ∈ 𝒫 (1...𝑁)(𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠))))))
28	9, 27	mpd 15	. 2 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 (1...𝑁)(𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)))))
29		r19.43 3102	. 2 ⊢ (∃𝑠 ∈ 𝒫 (1...𝑁)((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)))) ↔ (∃𝑠 ∈ 𝒫 (1...𝑁)(𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)))))
30	28, 29	sylibr 234	1 ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2113 ∃wrex 3058 {crab 3397 𝒫 cpw 4552 ⟨cop 4584 class class class wbr 5096 ↦ cmpt 5177 ^◡ccnv 5621 ↾ cres 5624 “ cima 5625 –1-1→wf1 6487 ‘cfv 6490 Isom wiso 6491 (class class class)co 7356 supcsup 9341 ℝcr 11023 1c1 11025 · cmul 11029 < clt 11164 ≤ cle 11165 − cmin 11362 ℕcn 12143 ...cfz 13421 ♯chash 14251

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-oadd 8399 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-sup 9343 df-dju 9811 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-n0 12400 df-xnn0 12473 df-z 12487 df-uz 12750 df-fz 13422 df-hash 14252

This theorem is referenced by: erdsze 35345

W3C validator