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Mirrors > Home > MPE Home > Th. List > Mathboxes > ccatcan2d | Structured version Visualization version GIF version |
Description: Cancellation law for concatenation. (Contributed by SN, 6-Sep-2023.) |
Ref | Expression |
---|---|
ccatcan2d.a | ⊢ (𝜑 → 𝐴 ∈ Word 𝑉) |
ccatcan2d.b | ⊢ (𝜑 → 𝐵 ∈ Word 𝑉) |
ccatcan2d.c | ⊢ (𝜑 → 𝐶 ∈ Word 𝑉) |
Ref | Expression |
---|---|
ccatcan2d | ⊢ (𝜑 → ((𝐴 ++ 𝐶) = (𝐵 ++ 𝐶) ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ++ 𝐶) = (𝐵 ++ 𝐶)) → (𝐴 ++ 𝐶) = (𝐵 ++ 𝐶)) | |
2 | ccatcan2d.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ Word 𝑉) | |
3 | lencl 13878 | . . . . . . . . 9 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0) | |
4 | 2, 3 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ0) |
5 | 4 | nn0cnd 11951 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐴) ∈ ℂ) |
6 | 5 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ++ 𝐶) = (𝐵 ++ 𝐶)) → (♯‘𝐴) ∈ ℂ) |
7 | ccatcan2d.b | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ Word 𝑉) | |
8 | lencl 13878 | . . . . . . . . 9 ⊢ (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℕ0) | |
9 | 7, 8 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ0) |
10 | 9 | nn0cnd 11951 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐵) ∈ ℂ) |
11 | 10 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ++ 𝐶) = (𝐵 ++ 𝐶)) → (♯‘𝐵) ∈ ℂ) |
12 | ccatcan2d.c | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Word 𝑉) | |
13 | lencl 13878 | . . . . . . . . 9 ⊢ (𝐶 ∈ Word 𝑉 → (♯‘𝐶) ∈ ℕ0) | |
14 | 12, 13 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (♯‘𝐶) ∈ ℕ0) |
15 | 14 | nn0cnd 11951 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐶) ∈ ℂ) |
16 | 15 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ++ 𝐶) = (𝐵 ++ 𝐶)) → (♯‘𝐶) ∈ ℂ) |
17 | ccatlen 13922 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐶 ∈ Word 𝑉) → (♯‘(𝐴 ++ 𝐶)) = ((♯‘𝐴) + (♯‘𝐶))) | |
18 | 2, 12, 17 | syl2anc 586 | . . . . . . . 8 ⊢ (𝜑 → (♯‘(𝐴 ++ 𝐶)) = ((♯‘𝐴) + (♯‘𝐶))) |
19 | fveq2 6663 | . . . . . . . 8 ⊢ ((𝐴 ++ 𝐶) = (𝐵 ++ 𝐶) → (♯‘(𝐴 ++ 𝐶)) = (♯‘(𝐵 ++ 𝐶))) | |
20 | 18, 19 | sylan9req 2876 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ++ 𝐶) = (𝐵 ++ 𝐶)) → ((♯‘𝐴) + (♯‘𝐶)) = (♯‘(𝐵 ++ 𝐶))) |
21 | ccatlen 13922 | . . . . . . . . 9 ⊢ ((𝐵 ∈ Word 𝑉 ∧ 𝐶 ∈ Word 𝑉) → (♯‘(𝐵 ++ 𝐶)) = ((♯‘𝐵) + (♯‘𝐶))) | |
22 | 7, 12, 21 | syl2anc 586 | . . . . . . . 8 ⊢ (𝜑 → (♯‘(𝐵 ++ 𝐶)) = ((♯‘𝐵) + (♯‘𝐶))) |
23 | 22 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ++ 𝐶) = (𝐵 ++ 𝐶)) → (♯‘(𝐵 ++ 𝐶)) = ((♯‘𝐵) + (♯‘𝐶))) |
24 | 20, 23 | eqtrd 2855 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ++ 𝐶) = (𝐵 ++ 𝐶)) → ((♯‘𝐴) + (♯‘𝐶)) = ((♯‘𝐵) + (♯‘𝐶))) |
25 | 6, 11, 16, 24 | addcan2ad 10839 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ++ 𝐶) = (𝐵 ++ 𝐶)) → (♯‘𝐴) = (♯‘𝐵)) |
26 | 1, 25 | oveq12d 7167 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ++ 𝐶) = (𝐵 ++ 𝐶)) → ((𝐴 ++ 𝐶) prefix (♯‘𝐴)) = ((𝐵 ++ 𝐶) prefix (♯‘𝐵))) |
27 | 26 | ex 415 | . . 3 ⊢ (𝜑 → ((𝐴 ++ 𝐶) = (𝐵 ++ 𝐶) → ((𝐴 ++ 𝐶) prefix (♯‘𝐴)) = ((𝐵 ++ 𝐶) prefix (♯‘𝐵)))) |
28 | pfxccat1 14059 | . . . . 5 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐶 ∈ Word 𝑉) → ((𝐴 ++ 𝐶) prefix (♯‘𝐴)) = 𝐴) | |
29 | 2, 12, 28 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ((𝐴 ++ 𝐶) prefix (♯‘𝐴)) = 𝐴) |
30 | pfxccat1 14059 | . . . . 5 ⊢ ((𝐵 ∈ Word 𝑉 ∧ 𝐶 ∈ Word 𝑉) → ((𝐵 ++ 𝐶) prefix (♯‘𝐵)) = 𝐵) | |
31 | 7, 12, 30 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ((𝐵 ++ 𝐶) prefix (♯‘𝐵)) = 𝐵) |
32 | 29, 31 | eqeq12d 2836 | . . 3 ⊢ (𝜑 → (((𝐴 ++ 𝐶) prefix (♯‘𝐴)) = ((𝐵 ++ 𝐶) prefix (♯‘𝐵)) ↔ 𝐴 = 𝐵)) |
33 | 27, 32 | sylibd 241 | . 2 ⊢ (𝜑 → ((𝐴 ++ 𝐶) = (𝐵 ++ 𝐶) → 𝐴 = 𝐵)) |
34 | oveq1 7156 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ++ 𝐶) = (𝐵 ++ 𝐶)) | |
35 | 33, 34 | impbid1 227 | 1 ⊢ (𝜑 → ((𝐴 ++ 𝐶) = (𝐵 ++ 𝐶) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 (class class class)co 7149 ℂcc 10528 + caddc 10533 ℕ0cn0 11891 ♯chash 13687 Word cword 13858 ++ cconcat 13917 prefix cpfx 14027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12890 df-fzo 13031 df-hash 13688 df-word 13859 df-concat 13918 df-substr 13998 df-pfx 14028 |
This theorem is referenced by: (None) |
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