ruclem30 - Metamath Proof Explorer

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Theorem ruclem30 7540

Description: Lemma for ruc 7550. A helper lemma for ruclem32 7542.

**Hypotheses**
Ref	Expression
ruclem.0	⊢ F:ℕ–→ℝ
ruclem.1	⊢ C = ({⟨1, ⟨((F ‘1) + 1), ((F ‘1) + 2)⟩⟩} ∪ (F ↾ (ℕ ∖ {1})))
ruclem.2	⊢ D = {⟨⟨x, y⟩, z⟩∣((x ∈ (ℝ × ℝ) ⋀ y ∈ ℝ) ⋀ z = if(((1^st ‘x) < y ⋀ y < (2^nd ‘x)), ⟨(((2 · y) + (2^nd ‘x)) / 3), ((y + (2 · (2^nd ‘x))) / 3)⟩, ⟨(((2 · (1^st ‘x)) + (2^nd ‘x)) / 3), (((1^st ‘x) + (2 · (2^nd ‘x))) / 3)⟩))}
ruclem.3	⊢ G = (1^st ∘ (Dseq₁C))
ruclem.4	⊢ H = (2^nd ∘ (Dseq₁C))
ruclem28.a	⊢ A ∈ ℕ
ruclem.b	⊢ B ∈ ℕ

**Assertion**
Ref	Expression
ruclem30	⊢ ((G ‘A) < (G ‘(A + B)) → (G ‘A) < (G ‘(A + (B + 1))))

Distinct variable groups: x,y,z z,F

**Proof of Theorem ruclem30**
Step	Hyp	Ref	Expression
1		ruclem.0	. . . 4 ⊢ F:ℕ–→ℝ
2		ruclem.1	. . . 4 ⊢ C = ({⟨1, ⟨((F ‘1) + 1), ((F ‘1) + 2)⟩⟩} ∪ (F ↾ (ℕ ∖ {1})))
3		ruclem.2	. . . 4 ⊢ D = {⟨⟨x, y⟩, z⟩∣((x ∈ (ℝ × ℝ) ⋀ y ∈ ℝ) ⋀ z = if(((1^st ‘x) < y ⋀ y < (2^nd ‘x)), ⟨(((2 · y) + (2^nd ‘x)) / 3), ((y + (2 · (2^nd ‘x))) / 3)⟩, ⟨(((2 · (1^st ‘x)) + (2^nd ‘x)) / 3), (((1^st ‘x) + (2 · (2^nd ‘x))) / 3)⟩))}
4		ruclem.3	. . . 4 ⊢ G = (1^st ∘ (Dseq₁C))
5		ruclem.4	. . . 4 ⊢ H = (2^nd ∘ (Dseq₁C))
6		ruclem28.a	. . . . 5 ⊢ A ∈ ℕ
7		ruclem.b	. . . . 5 ⊢ B ∈ ℕ
8		nnaddclt 5942	. . . . 5 ⊢ ((A ∈ ℕ ⋀ B ∈ ℕ) → (A + B) ∈ ℕ)
9	6, 7, 8	mp2an 699	. . . 4 ⊢ (A + B) ∈ ℕ
10	1, 2, 3, 4, 5, 9	ruclem26 7536	. . 3 ⊢ (G ‘(A + B)) < (G ‘((A + B) + 1))
11	6	nncn 5934	. . . . 5 ⊢ A ∈ ℂ
12	7	nncn 5934	. . . . 5 ⊢ B ∈ ℂ
13		ax1cn 5281	. . . . 5 ⊢ 1 ∈ ℂ
14	11, 12, 13	addass 5336	. . . 4 ⊢ ((A + B) + 1) = (A + (B + 1))
15	14	fveq2i 3733	. . 3 ⊢ (G ‘((A + B) + 1)) = (G ‘(A + (B + 1)))
16	10, 15	breqtr 2643	. 2 ⊢ (G ‘(A + B)) < (G ‘(A + (B + 1)))
17	1, 2, 3, 4, 5, 6	ruclem22 7532	. . 3 ⊢ (G ‘A) ∈ ℝ
18	1, 2, 3, 4, 5, 9	ruclem22 7532	. . 3 ⊢ (G ‘(A + B)) ∈ ℝ
19		peano2nn 5937	. . . . . 6 ⊢ (B ∈ ℕ → (B + 1) ∈ ℕ)
20	7, 19	ax-mp 7	. . . . 5 ⊢ (B + 1) ∈ ℕ
21		nnaddclt 5942	. . . . 5 ⊢ ((A ∈ ℕ ⋀ (B + 1) ∈ ℕ) → (A + (B + 1)) ∈ ℕ)
22	6, 20, 21	mp2an 699	. . . 4 ⊢ (A + (B + 1)) ∈ ℕ
23	1, 2, 3, 4, 5, 22	ruclem22 7532	. . 3 ⊢ (G ‘(A + (B + 1))) ∈ ℝ
24	17, 18, 23	lttr 5597	. 2 ⊢ (((G ‘A) < (G ‘(A + B)) ⋀ (G ‘(A + B)) < (G ‘(A + (B + 1)))) → (G ‘A) < (G ‘(A + (B + 1))))
25	16, 24	mpan2 698	1 ⊢ ((G ‘A) < (G ‘(A + B)) → (G ‘A) < (G ‘(A + (B + 1))))

Colors of variables: wff set class

Syntax hints: → wi 3 ⋀ wa 223 = wceq 958 ∈ wcel 960 ∖ cdif 2047 ∪ cun 2048 ifcif 2365 {csn 2413 ⟨cop 2415 class class class wbr 2624 × cxp 3174 ↾ cres 3178 ∘ ccom 3180 –→wf 3184 ‘cfv 3188 (class class class)co 3969 {copab2 3970 1^st c1st 4083 2^nd c2nd 4084 ℝcr 5245 1c1 5247 + caddc 5249 · cmul 5251 / cdiv 5306 ℕcn 5308 < clt 5498 2c2 5963 3c3 5964 seq₁cseq1 6308

This theorem is referenced by: ruclem32 7542

This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 964 ax-gen 965 ax-8 966 ax-9 967 ax-10 968 ax-11 969 ax-12 970 ax-13 971 ax-14 972 ax-17 973 ax-4 975 ax-5o 977 ax-6o 980 ax-9o 1125 ax-10o 1142 ax-16 1212 ax-11o 1220 ax-ext 1462 ax-rep 2698 ax-sep 2708 ax-nul 2715 ax-pow 2748 ax-pr 2785 ax-un 2872 ax-inf2 4634

This theorem depends on definitions: df-bi 147 df-or 224 df-an 225 df-3or 778 df-3an 779 df-ex 983 df-sb 1174 df-eu 1384 df-mo 1385 df-clab 1467 df-cleq 1472 df-clel 1475 df-ne 1590 df-nel 1591 df-ral 1652 df-rex 1653 df-reu 1654 df-rab 1655 df-v 1815 df-sbc 1945 df-csb 2005 df-dif 2052 df-un 2053 df-in 2054 df-ss 2056 df-pss 2058 df-nul 2284 df-if 2366 df-pw 2406 df-sn 2416 df-pr 2417 df-tp 2419 df-op 2420 df-uni 2508 df-int 2538 df-iun 2572 df-br 2625 df-opab 2672 df-tr 2686 df-eprel 2838 df-id 2841 df-po 2846 df-so 2856 df-fr 2923 df-we 2940 df-ord 2957 df-on 2958 df-lim 2959 df-suc 2960 df-om 3138 df-xp 3190 df-rel 3191 df-cnv 3192 df-co 3193 df-dm 3194 df-rn 3195 df-res 3196 df-ima 3197 df-fun 3198 df-fn 3199 df-f 3200 df-f1 3201 df-fo 3202 df-f1o 3203 df-fv 3204 df-rdg 3938 df-opr 3971 df-oprab 3972 df-1st 4085 df-2nd 4086 df-1o 4139 df-oadd 4141 df-omul 4142 df-er 4267 df-ec 4269 df-qs 4272 df-en 4374 df-dom 4375 df-sdom 4376 df-ni 5012 df-pli 5013 df-mi 5014 df-lti 5015 df-plpq 5047 df-mpq 5048 df-enq 5049 df-nq 5050 df-plq 5051 df-mq 5052 df-rq 5053 df-ltq 5054 df-1q 5055 df-np 5098 df-1p 5099 df-plp 5100 df-mp 5101 df-ltp 5102 df-plpr 5176 df-mpr 5177 df-enr 5178 df-nr 5179 df-plr 5180 df-mr 5181 df-ltr 5182 df-0r 5183 df-1r 5184 df-m1r 5185 df-c 5252 df-0 5253 df-1 5254 df-i 5255 df-r 5256 df-plus 5257 df-mul 5258 df-lt 5259 df-sub 5368 df-neg 5370 df-pnf 5499 df-mnf 5500 df-xr 5501 df-ltxr 5502 df-le 5503 df-div 5715 df-n 5927 df-2 5972 df-3 5973 df-n0 6102 df-z 6138 df-seq1 6309