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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > chnlt | Structured version Visualization version GIF version |
Description: Compare any two elements in a chain. (Contributed by Thierry Arnoux, 19-Jun-2025.) |
Ref | Expression |
---|---|
chnlt.1 | ⊢ (𝜑 → < Po 𝐴) |
chnlt.2 | ⊢ (𝜑 → 𝐶 ∈ ( < Chain𝐴)) |
chnlt.3 | ⊢ (𝜑 → 𝐽 ∈ (0..^(♯‘𝐶))) |
chnlt.4 | ⊢ (𝜑 → 𝐼 ∈ (0..^𝐽)) |
Ref | Expression |
---|---|
chnlt | ⊢ (𝜑 → (𝐶‘𝐼) < (𝐶‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chnlt.1 | . . 3 ⊢ (𝜑 → < Po 𝐴) | |
2 | chnlt.2 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ( < Chain𝐴)) | |
3 | chnlt.3 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ (0..^(♯‘𝐶))) | |
4 | fzofzp1 13800 | . . . . 5 ⊢ (𝐽 ∈ (0..^(♯‘𝐶)) → (𝐽 + 1) ∈ (0...(♯‘𝐶))) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐽 + 1) ∈ (0...(♯‘𝐶))) |
6 | 2, 5 | pfxchn 32984 | . . 3 ⊢ (𝜑 → (𝐶 prefix (𝐽 + 1)) ∈ ( < Chain𝐴)) |
7 | chnlt.4 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ (0..^𝐽)) | |
8 | fzossz 13716 | . . . . . . . 8 ⊢ (0..^(♯‘𝐶)) ⊆ ℤ | |
9 | 8, 3 | sselid 3993 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ) |
10 | 9 | zcnd 12721 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℂ) |
11 | 1cnd 11254 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ) | |
12 | 2 | chnwrd 32982 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Word 𝐴) |
13 | pfxlen 14718 | . . . . . . 7 ⊢ ((𝐶 ∈ Word 𝐴 ∧ (𝐽 + 1) ∈ (0...(♯‘𝐶))) → (♯‘(𝐶 prefix (𝐽 + 1))) = (𝐽 + 1)) | |
14 | 12, 5, 13 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (♯‘(𝐶 prefix (𝐽 + 1))) = (𝐽 + 1)) |
15 | 10, 11, 14 | mvrraddd 11673 | . . . . 5 ⊢ (𝜑 → ((♯‘(𝐶 prefix (𝐽 + 1))) − 1) = 𝐽) |
16 | 15 | oveq2d 7447 | . . . 4 ⊢ (𝜑 → (0..^((♯‘(𝐶 prefix (𝐽 + 1))) − 1)) = (0..^𝐽)) |
17 | 7, 16 | eleqtrrd 2842 | . . 3 ⊢ (𝜑 → 𝐼 ∈ (0..^((♯‘(𝐶 prefix (𝐽 + 1))) − 1))) |
18 | 1, 6, 17 | chnub 32986 | . 2 ⊢ (𝜑 → ((𝐶 prefix (𝐽 + 1))‘𝐼) < (lastS‘(𝐶 prefix (𝐽 + 1)))) |
19 | fzo0ssnn0 13782 | . . . . . 6 ⊢ (0..^(♯‘𝐶)) ⊆ ℕ0 | |
20 | 19, 3 | sselid 3993 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ ℕ0) |
21 | fzossfzop1 13779 | . . . . 5 ⊢ (𝐽 ∈ ℕ0 → (0..^𝐽) ⊆ (0..^(𝐽 + 1))) | |
22 | 20, 21 | syl 17 | . . . 4 ⊢ (𝜑 → (0..^𝐽) ⊆ (0..^(𝐽 + 1))) |
23 | 22, 7 | sseldd 3996 | . . 3 ⊢ (𝜑 → 𝐼 ∈ (0..^(𝐽 + 1))) |
24 | pfxfv 14717 | . . 3 ⊢ ((𝐶 ∈ Word 𝐴 ∧ (𝐽 + 1) ∈ (0...(♯‘𝐶)) ∧ 𝐼 ∈ (0..^(𝐽 + 1))) → ((𝐶 prefix (𝐽 + 1))‘𝐼) = (𝐶‘𝐼)) | |
25 | 12, 5, 23, 24 | syl3anc 1370 | . 2 ⊢ (𝜑 → ((𝐶 prefix (𝐽 + 1))‘𝐼) = (𝐶‘𝐼)) |
26 | lencl 14568 | . . . . . 6 ⊢ (𝐶 ∈ Word 𝐴 → (♯‘𝐶) ∈ ℕ0) | |
27 | 12, 26 | syl 17 | . . . . 5 ⊢ (𝜑 → (♯‘𝐶) ∈ ℕ0) |
28 | fz0add1fz1 13771 | . . . . 5 ⊢ (((♯‘𝐶) ∈ ℕ0 ∧ 𝐽 ∈ (0..^(♯‘𝐶))) → (𝐽 + 1) ∈ (1...(♯‘𝐶))) | |
29 | 27, 3, 28 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐽 + 1) ∈ (1...(♯‘𝐶))) |
30 | pfxfvlsw 14730 | . . . 4 ⊢ ((𝐶 ∈ Word 𝐴 ∧ (𝐽 + 1) ∈ (1...(♯‘𝐶))) → (lastS‘(𝐶 prefix (𝐽 + 1))) = (𝐶‘((𝐽 + 1) − 1))) | |
31 | 12, 29, 30 | syl2anc 584 | . . 3 ⊢ (𝜑 → (lastS‘(𝐶 prefix (𝐽 + 1))) = (𝐶‘((𝐽 + 1) − 1))) |
32 | 10, 11 | pncand 11619 | . . . 4 ⊢ (𝜑 → ((𝐽 + 1) − 1) = 𝐽) |
33 | 32 | fveq2d 6911 | . . 3 ⊢ (𝜑 → (𝐶‘((𝐽 + 1) − 1)) = (𝐶‘𝐽)) |
34 | 31, 33 | eqtrd 2775 | . 2 ⊢ (𝜑 → (lastS‘(𝐶 prefix (𝐽 + 1))) = (𝐶‘𝐽)) |
35 | 18, 25, 34 | 3brtr3d 5179 | 1 ⊢ (𝜑 → (𝐶‘𝐼) < (𝐶‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 class class class wbr 5148 Po wpo 5595 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 + caddc 11156 − cmin 11490 ℕ0cn0 12524 ℤcz 12611 ...cfz 13544 ..^cfzo 13691 ♯chash 14366 Word cword 14549 lastSclsw 14597 prefix cpfx 14705 Chaincchn 32979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-lsw 14598 df-concat 14606 df-s1 14631 df-substr 14676 df-pfx 14706 df-chn 32980 |
This theorem is referenced by: chnso 32988 |
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