nnf1o - Intuitionistic Logic Explorer

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Theorem nnf1o 12143

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 9671	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
2		nnf1o.mn	. . . . . 6 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
3	2	simprd 114	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4	3	nnzd 9767	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
5	1, 4	fzfigd 10868	. . 3 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
6		nnf1o.m	. . . . 5 ⊢ (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)
7		f1ocnv 5652	. . . . 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 5662	. . . 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 415	. . 3 ⊢ (𝜑 → (^◡𝐹 ∘ 𝐺):(1...𝑁)–1-1-onto→(1...𝑀))
12	5, 11	fihasheqf1od 11228	. 2 ⊢ (𝜑 → (♯‘(1...𝑁)) = (♯‘(1...𝑀)))
13		nnnn0 9570	. . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
14		hashfz1 11222	. . 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 9570	. . 3 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
18		hashfz1 11222	. . 3 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
19	16, 17, 18	3syl 17	. 2 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
20	12, 15, 19	3eqtr3d 2279	1 ⊢ (𝜑 → 𝑁 = 𝑀)

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 = wceq 1402 ∈ wcel 2209 ^◡ccnv 4773 ∘ ccom 4778 –1-1-onto→wf1o 5376 ‘cfv 5377 (class class class)co 6085 1c1 8180 ℕcn 9304 ℕ₀cn0 9563 ...cfz 10411 ♯chash 11214

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-addcom 8279 ax-addass 8281 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-0id 8287 ax-rnegex 8288 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-1o 6687 df-er 6807 df-en 7023 df-dom 7024 df-fin 7025 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-inn 9305 df-n0 9564 df-z 9645 df-uz 9922 df-fz 10412 df-ihash 11215

This theorem is used by: summodclem3 12147 prodmodclem3 12342

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