nnf1o - Intuitionistic Logic Explorer

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Theorem nnf1o 11902

Description: Lemma for sum and product theorems. (Contributed by Jim Kingdon, 15-Aug-2022.)

**Hypotheses**
Ref	Expression
nnf1o.mn	⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
nnf1o.m	⊢ (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)
nnf1o.n	⊢ (𝜑 → 𝐺:(1...𝑁)–1-1-onto→𝐴)

**Assertion**
Ref	Expression
nnf1o	⊢ (𝜑 → 𝑁 = 𝑀)

**Proof of Theorem nnf1o**
Step	Hyp	Ref	Expression
1		1zzd 9484	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
2		nnf1o.mn	. . . . . 6 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
3	2	simprd 114	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4	3	nnzd 9579	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
5	1, 4	fzfigd 10665	. . 3 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
6		nnf1o.m	. . . . 5 ⊢ (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)
7		f1ocnv 5587	. . . . 5 ⊢ (𝐹:(1...𝑀)–1-1-onto→𝐴 → ^◡𝐹:𝐴–1-1-onto→(1...𝑀))
8	6, 7	syl 14	. . . 4 ⊢ (𝜑 → ^◡𝐹:𝐴–1-1-onto→(1...𝑀))
9		nnf1o.n	. . . 4 ⊢ (𝜑 → 𝐺:(1...𝑁)–1-1-onto→𝐴)
10		f1oco 5597	. . . 4 ⊢ ((^◡𝐹:𝐴–1-1-onto→(1...𝑀) ∧ 𝐺:(1...𝑁)–1-1-onto→𝐴) → (^◡𝐹 ∘ 𝐺):(1...𝑁)–1-1-onto→(1...𝑀))
11	8, 9, 10	syl2anc 411	. . 3 ⊢ (𝜑 → (^◡𝐹 ∘ 𝐺):(1...𝑁)–1-1-onto→(1...𝑀))
12	5, 11	fihasheqf1od 11023	. 2 ⊢ (𝜑 → (♯‘(1...𝑁)) = (♯‘(1...𝑀)))
13		nnnn0 9387	. . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
14		hashfz1 11017	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
15	3, 13, 14	3syl 17	. 2 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
16	2	simpld 112	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ)
17		nnnn0 9387	. . 3 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
18		hashfz1 11017	. . 3 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
19	16, 17, 18	3syl 17	. 2 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
20	12, 15, 19	3eqtr3d 2270	1 ⊢ (𝜑 → 𝑁 = 𝑀)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1395 ∈ wcel 2200 ^◡ccnv 4718 ∘ ccom 4723 –1-1-onto→wf1o 5317 ‘cfv 5318 (class class class)co 6007 1c1 8011 ℕcn 9121 ℕ₀cn0 9380 ...cfz 10216 ♯chash 11009

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-addcom 8110 ax-addass 8112 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-0id 8118 ax-rnegex 8119 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-apti 8125 ax-pre-ltadd 8126

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-frec 6543 df-1o 6568 df-er 6688 df-en 6896 df-dom 6897 df-fin 6898 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-inn 9122 df-n0 9381 df-z 9458 df-uz 9734 df-fz 10217 df-ihash 11010

This theorem is referenced by: summodclem3 11906 prodmodclem3 12101

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