erdszelem11 - Mathbox for Mario Carneiro

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Theorem erdszelem11 35169

Description: Lemma for erdsze 35170. (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 35168	. . 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 11370	. . . . . . 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 35165	. . . . . 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 11371	. . . . . . 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 35165	. . . . . 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 965	. . . 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 3161	. . 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 3128	. 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 846 = wceq 1537 ∈ wcel 2108 ∃wrex 3076 {crab 3443 𝒫 cpw 4622 ⟨cop 4654 class class class wbr 5166 ↦ cmpt 5249 ^◡ccnv 5699 ↾ cres 5702 “ cima 5703 –1-1→wf1 6570 ‘cfv 6573 Isom wiso 6574 (class class class)co 7448 supcsup 9509 ℝcr 11183 1c1 11185 · cmul 11189 < clt 11324 ≤ cle 11325 − cmin 11520 ℕcn 12293 ...cfz 13567 ♯chash 14379

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-oadd 8526 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-n0 12554 df-xnn0 12626 df-z 12640 df-uz 12904 df-fz 13568 df-hash 14380

This theorem is referenced by: erdsze 35170

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