erdszelem11 - Mathbox for Mario Carneiro

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Theorem erdszelem11 35207

Description: Lemma for erdsze 35208. (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 35206	. . 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 11342	. . . . . . 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 35203	. . . . . 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 11343	. . . . . . 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 35203	. . . . . 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 3154	. . 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 3121	. 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 1539 ∈ wcel 2107 ∃wrex 3069 {crab 3435 𝒫 cpw 4599 ⟨cop 4631 class class class wbr 5142 ↦ cmpt 5224 ^◡ccnv 5683 ↾ cres 5686 “ cima 5687 –1-1→wf1 6557 ‘cfv 6560 Isom wiso 6561 (class class class)co 7432 supcsup 9481 ℝcr 11155 1c1 11157 · cmul 11161 < clt 11296 ≤ cle 11297 − cmin 11493 ℕcn 12267 ...cfz 13548 ♯chash 14370

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-oadd 8511 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-dju 9942 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-n0 12529 df-xnn0 12602 df-z 12616 df-uz 12880 df-fz 13549 df-hash 14371

This theorem is referenced by: erdsze 35208

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