uzind2 - Metamath Proof Explorer

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Theorem uzind2 6208

Description: Induction on the upper integers that start after an integer M. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction hypothesis.

**Hypotheses**
Ref	Expression
uzind2.1	⊢ (j = (M + 1) → (φ ↔ ψ))
uzind2.2	⊢ (j = k → (φ ↔ χ))
uzind2.3	⊢ (j = (k + 1) → (φ ↔ θ))
uzind2.4	⊢ (j = N → (φ ↔ τ))
uzind2.5	⊢ (M ∈ ℤ → ψ)
uzind2.6	⊢ ((M ∈ ℤ ⋀ k ∈ ℤ ⋀ M < k) → (χ → θ))

**Assertion**
Ref	Expression
uzind2	⊢ ((M ∈ ℤ ⋀ N ∈ ℤ ⋀ M < N) → τ)

Distinct variable groups: j,N ψ,j χ,j θ,j τ,j φ,k j,k,M

**Proof of Theorem uzind2**
Step	Hyp	Ref	Expression
1		zltp1let 6183	. . 3 ⊢ ((M ∈ ℤ ⋀ N ∈ ℤ) → (M < N ↔ (M + 1) ≤ N))
2		peano2z 6168	. . . . . . 7 ⊢ (M ∈ ℤ → (M + 1) ∈ ℤ)
3		uzind2.1	. . . . . . . . . 10 ⊢ (j = (M + 1) → (φ ↔ ψ))
4	3	imbi2d 614	. . . . . . . . 9 ⊢ (j = (M + 1) → ((M ∈ ℤ → φ) ↔ (M ∈ ℤ → ψ)))
5		uzind2.2	. . . . . . . . . 10 ⊢ (j = k → (φ ↔ χ))
6	5	imbi2d 614	. . . . . . . . 9 ⊢ (j = k → ((M ∈ ℤ → φ) ↔ (M ∈ ℤ → χ)))
7		uzind2.3	. . . . . . . . . 10 ⊢ (j = (k + 1) → (φ ↔ θ))
8	7	imbi2d 614	. . . . . . . . 9 ⊢ (j = (k + 1) → ((M ∈ ℤ → φ) ↔ (M ∈ ℤ → θ)))
9		uzind2.4	. . . . . . . . . 10 ⊢ (j = N → (φ ↔ τ))
10	9	imbi2d 614	. . . . . . . . 9 ⊢ (j = N → ((M ∈ ℤ → φ) ↔ (M ∈ ℤ → τ)))
11		uzind2.5	. . . . . . . . . 10 ⊢ (M ∈ ℤ → ψ)
12	11	a1i 8	. . . . . . . . 9 ⊢ ((M + 1) ∈ ℤ → (M ∈ ℤ → ψ))
13		zltp1let 6183	. . . . . . . . . . . . . . 15 ⊢ ((M ∈ ℤ ⋀ k ∈ ℤ) → (M < k ↔ (M + 1) ≤ k))
14		uzind2.6	. . . . . . . . . . . . . . . 16 ⊢ ((M ∈ ℤ ⋀ k ∈ ℤ ⋀ M < k) → (χ → θ))
15	14	3expia 837	. . . . . . . . . . . . . . 15 ⊢ ((M ∈ ℤ ⋀ k ∈ ℤ) → (M < k → (χ → θ)))
16	13, 15	sylbird 205	. . . . . . . . . . . . . 14 ⊢ ((M ∈ ℤ ⋀ k ∈ ℤ) → ((M + 1) ≤ k → (χ → θ)))
17	16	ex 373	. . . . . . . . . . . . 13 ⊢ (M ∈ ℤ → (k ∈ ℤ → ((M + 1) ≤ k → (χ → θ))))
18	17	com3l 34	. . . . . . . . . . . 12 ⊢ (k ∈ ℤ → ((M + 1) ≤ k → (M ∈ ℤ → (χ → θ))))
19	18	imp 350	. . . . . . . . . . 11 ⊢ ((k ∈ ℤ ⋀ (M + 1) ≤ k) → (M ∈ ℤ → (χ → θ)))
20	19	3adant1 799	. . . . . . . . . 10 ⊢ (((M + 1) ∈ ℤ ⋀ k ∈ ℤ ⋀ (M + 1) ≤ k) → (M ∈ ℤ → (χ → θ)))
21	20	a2d 13	. . . . . . . . 9 ⊢ (((M + 1) ∈ ℤ ⋀ k ∈ ℤ ⋀ (M + 1) ≤ k) → ((M ∈ ℤ → χ) → (M ∈ ℤ → θ)))
22	4, 6, 8, 10, 12, 21	uzind 6207	. . . . . . . 8 ⊢ (((M + 1) ∈ ℤ ⋀ N ∈ ℤ ⋀ (M + 1) ≤ N) → (M ∈ ℤ → τ))
23	22	3exp 834	. . . . . . 7 ⊢ ((M + 1) ∈ ℤ → (N ∈ ℤ → ((M + 1) ≤ N → (M ∈ ℤ → τ))))
24	2, 23	syl 10	. . . . . 6 ⊢ (M ∈ ℤ → (N ∈ ℤ → ((M + 1) ≤ N → (M ∈ ℤ → τ))))
25	24	com34 36	. . . . 5 ⊢ (M ∈ ℤ → (N ∈ ℤ → (M ∈ ℤ → ((M + 1) ≤ N → τ))))
26	25	pm2.43a 66	. . . 4 ⊢ (M ∈ ℤ → (N ∈ ℤ → ((M + 1) ≤ N → τ)))
27	26	imp 350	. . 3 ⊢ ((M ∈ ℤ ⋀ N ∈ ℤ) → ((M + 1) ≤ N → τ))
28	1, 27	sylbid 203	. 2 ⊢ ((M ∈ ℤ ⋀ N ∈ ℤ) → (M < N → τ))
29	28	3impia 832	1 ⊢ ((M ∈ ℤ ⋀ N ∈ ℤ ⋀ M < N) → τ)

Colors of variables: wff set class

Syntax hints: → wi 3 ↔ wb 146 ⋀ wa 223 ⋀ w3a 777 = wceq 958 ∈ wcel 960 class class class wbr 2624 (class class class)co 3969 1c1 5247 + caddc 5249 ≤ cle 5307 ℤcz 5310 < clt 5498

This theorem is referenced by: fsum1ps 7018

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-n 5927 df-n0 6102 df-z 6138