nnf1o - Intuitionistic Logic Explorer

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Theorem nnf1o 12000

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 9550	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
2		nnf1o.mn	. . . . . 6 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
3	2	simprd 114	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4	3	nnzd 9645	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
5	1, 4	fzfigd 10739	. . 3 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
6		nnf1o.m	. . . . 5 ⊢ (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)
7		f1ocnv 5605	. . . . 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 5615	. . . 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 11097	. 2 ⊢ (𝜑 → (♯‘(1...𝑁)) = (♯‘(1...𝑀)))
13		nnnn0 9451	. . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
14		hashfz1 11091	. . 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 9451	. . 3 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
18		hashfz1 11091	. . 3 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
19	16, 17, 18	3syl 17	. 2 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
20	12, 15, 19	3eqtr3d 2272	1 ⊢ (𝜑 → 𝑁 = 𝑀)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1398 ∈ wcel 2202 ^◡ccnv 4730 ∘ ccom 4735 –1-1-onto→wf1o 5332 ‘cfv 5333 (class class class)co 6028 1c1 8076 ℕcn 9185 ℕ₀cn0 9444 ...cfz 10288 ♯chash 11083

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-addcom 8175 ax-addass 8177 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-0id 8183 ax-rnegex 8184 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-frec 6600 df-1o 6625 df-er 6745 df-en 6953 df-dom 6954 df-fin 6955 df-pnf 8258 df-mnf 8259 df-xr 8260 df-ltxr 8261 df-le 8262 df-sub 8394 df-neg 8395 df-inn 9186 df-n0 9445 df-z 9524 df-uz 9800 df-fz 10289 df-ihash 11084

This theorem is referenced by: summodclem3 12004 prodmodclem3 12199

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